I'm a self-taught musician and have been playing for years, but being self-taught really limited my knowledge of music theory (in that I know pretty much nothing about it) and I'm having difficulty improvising because of it. My question, at its base would be, "How do scales, key, chords, and anything else I might have left out, relate back to each other and why are they useful as a whole?" Additionally, how might one go about approaching learning music theory without a teacher?
There are many, many different concepts in music theory and the goal of all of them is to understand what makes music sound the way it does, how music relates to other music, and utilize this knowledge on new material.
Not every theory concept can be used effectively on a given piece, for example if you tried to do Roman Numeral analysis on an atonal piece you'd be wasting your time as it is not tonal in nature. Also not all the terms are related in nature for example the meter of the piece (represented by a time signature) has nothing to do with the key of the piece (represented by the key signature), but they affect how the song will sound independently and there are pieces that don't use one or the other or both. Even in analysis, form analysis is independent from harmonic analysis, but they both give you vital information about the piece you are looking at.
Scales, keys, and chords are actually pretty intermingled as a key will imply a specific scale and harmony (set of chords) although a piece can use other chords or scales inside of it and be in the same key. Again as stated above all these terms are used to understand a piece of music
Just to make sure we are on the same page, I'll define each concept real quick:
A quick example is in the key of C major you'll most likely work with the C major scale and chords built off the C major scale which are used to imply that C is the tonic, however a piece may use other scales and chords that are from outside the scale and still be in the key of C or even modulate to a new key if desired which has its own implied tonic and harmony. These terms are all used in conjunction, but they describe different ideas.
Music theory is pretty dense if you don't have anyone to guide you as you'll need the feedback when you get into the more advanced topics. There are a lot of resources out there, but they all approach the material in a slightly different way and honestly, I've never found a theory book that I'm 100% on their terminology or how they use symbols. Musictheory.net is a very good introduction to the subject although the content is very shallow in nature. Even just going though the questions tagged theory on the site should give you a start.
Music is that stuff composed of air vibrations that sometimes makes our ears and brains happy.
Music Theory is a vast body of rules-of-thumb developed by humans over the centuries in an attempt to understand WHY a particular sequence of frequencies makes our ears happy. Music and (some) people's enjoyment of it comes first, and people liking a particular piece of Music always trumps any attempt by Theorists to impose rules on the process. The rules of Music Theory can always be broken, but taste and fashion in music cannot be dictated. The "rules" of Music Theory only exist in an attempt to inform would-be creators of music as to which groups of frequencies played sequentially or simultaneously are more likely to please some people's ears.
Frequencies are continuous. Every possible frequency of air vibration is a note... but almost all of those possible combinations will sound "out of tune" or "dissonant" to our ears. All frequencies playing at once is noise or cacophony -- not music. Music Theory includes the analysis of why we have selected only 12 frequencies and their power-of-2-multiples (various octaves), out of an infinite number of possible frequencies, on which to base our music. Music Theory can also inform you that 12 is not the only possible choice, and that the next 2 best choices would be 19 notes per octave, or 31 notes per octave. (The details here are entirely mathematical).
Of a given set of 12 notes, we rarely use all 12 (except for the case of atonal music and 12-tone music... both of limited popular appeal). Usually of the 12 notes, any given phrase of music will confine itself to only 7 (diatonic scale) or 5 (pentatonic scale) distinct notes.
Once we have chosen 7 (or 5) of the 12 possible notes, which one should you start and end on? This defines the Tonal Center, and is closely related to the concepts of Keys, Modes, and the distinction between Major and Minor Keys (which has mostly replaced Modal scales).
Before the 1700's, Music Theorists hadn't developed chord-theory as we recognize it today. Back then, the simultaneous notes c-e-g and g-c-e were recognized as 2 distinct chords. There were hundreds of chords (all possible permutations of 2, 3, or 4 notes at a time?), and a vast body of learning to help a composer to choose pleasing chord progressions. Jean-Philippe Rameau, in Treatise on Harmony (1722), "figured out" mathematically that c-e-g and g-c-e and even e-g-c (any combination of the notes c, and e, and g, in any octaves) were the same chord! In terms of 3 note chords, Rameau developed a "Theory" that replaced hundreds of chords with a much more manageable 24 chords (a major and a minor chord built on each of the 12 possible notes). Today, one cannot discuss what Western Music is doing without implicitly using Rameau's Theory of chords -- it has become like air: nearly everywhere, but nearly invisible.
Music Theory goes on and on. You can get as deep into it as you like, or you can just pick up bits and pieces over decades. It is stories and ideas and as much mathematics as you can stomach. It can help you if you are writing music. It can give you several different perspectives to answer the question of "why do THESE notes sound good?" If you are plinking at a piano or a guitar, and you do some cool riff, it is probably the fact that someone has already given a name to what you are doing -- not the riff itself, but "what is it about this twist that sounds good?" Once you have a name for a phenomenon, then you can apply it in other places.
Don't rush it. Don't force it. Keep playing music, always... and if you get a chance, crack a book.
A suggested sequence (depends a lot on what instruments you have available, but to learn Harmony and Chord-progressions, you really need access to at least a guitar or a keyboard).
Learn some chords on a guitar. You only need 3 chords (I-IV-V) to play over half of all songs. As a suggestion, learn (major) C, F, G, D, A, E, and (minor) Dm, Em, Am. Get those piano-guitar-vocals books with the little chord symbols to learn chords for songs you like. Just strum. Really listen to chord progressions.
Figure out which notes are in each chord. Look at the chord "shape" and figure out which note you are playing on each string. Eventually you'll have to memorize that a C (major) chord is composed of c-e-g, for each chord, but there's a pattern to it.
Learn the Circle-of-Fifths. This provides a pattern for the 12-major and 12-minor keys and their key-signatures (sharps & flats). Learn the keys. There are patterns, watch for them.
Memorize that "one more sharp" always proceeds in this pattern: f-c-g-d-a-e-b Memorize that "one more flat" always proceeds in this pattern: b-e-a-d-g-c-f (the reverse) So in a key with 3 sharps, the sharps will always be f-c-g. In a key with 4 flats, the flats will always be b-e-a-d.
Learn how to pick out the 3 basic notes on a keyboard for each of the 12-major and 12-minor chords. There are patterns going on here... so this is not as daunting as it sounds. You don't have to learn to really play keyboards, but it's a lot easier to see the notes of each chord on a keyboard. The chords make specific shapes on the keyboard. (for using a keyboard to study harmony this way, even just a cheap $20 piece of trash keyboard would be sufficient).
Learn the intervals, and how to translate steps of the scale to the notes. The scale of C-major is c-d-e-f-g-a-b-c Count from 1 to 7 against these notes (8=1 again). so c=1 d=2 e=3 f=4 g=5 a=6 b=7 c=8=1. If I ask for the 3rd in Cmajor, I'm talking about the 'e'. This is all relative to what scale or key we're in. In D-major, the scale is d=1, e=2, f#=3, g=4, a=5, b=6, c#=7, d=8=1. So in D-major, e is the 2nd, but in C-major, e is the 3rd. You get used to it.
Having learned Keys, Key-signatures, and maybe at least half of the major and minor chords (seriously, almost no one spends a lot of time in G-flat)... now add 7th's to your basic chords. From C-major, learn C7 & Cmaj7. From C-minor, learn Cmin7.
There are other chord permutations: diminished & augmented... ignore those until you decide to either take a whole course on Harmony, and/or to become a composer. You might have to play 1 or 2 for a specific song, but they are a relatively minor detail the grand-scheme of things. The same thing applies to 9th, 11th, and 13th chords... these are just tacking a few MORE notes onto one of the basic 7th chords. If you don't know what the 9th or 11th is... just leave it out, and you can still play the song.
Since you know the key-signatures, now learn the Scales. Start with the major scale, and the natural-minor scale (the one where the accidentals match the key-signature).
If you play guitar, or specifically the blues: learn the Blues scale.
Learn the pentatonic scales (a question of what to leave out).
Later learn the melodic-minor scale and the harmonic-minor scale.
Part of introductory musical instrument instruction is to play scales over and over and over. The purpose of this is to develop "muscle-memory" in your hands so that your fingers automatically go to the correct next note. This is important for performance and improvisation. For the sake of Music Theory, however, just learn scales so that you can figure out which notes belong to a particular type of scale in a particular key in less than a minute or two. ;)
Learn Roman-Numeral transposition... just learn what it means, not how to do it instantly like Studio Musicians. Look at the chord progressions for some of your favorite songs. If the song is in the Key of C, then "I"=C, "IV"=F, and "V"=G. If you know a song in one key, convert the chords to Roman-Numeral notation, and start paying attention to how often most songs spend 80% of their time on the chords I-IV-V. In this notation, the particular Key becomes irrelevant, and you can start studying Harmonic/Chord Progressions (stripping away the details to focus on what's going on at a much more generic level).
To improvise, you first should know the chord progression. At some moment in time, you will be in a particular chord. The notes of that chord are the notes that it is safer to dwell upon.
Example: We're playing Blues in the key of E-major. In the Blues, we almost always use the 7th chords, so I-->I7, IV-->IV7, V-->V7. I7=E7, IV=A7, V=B7. The progression for a 12-bar-blues might be: I7 IV7 I7 I7 IV7 IV7 I7 I7 V7 IV7 I7 V7 In E-major, this translates to: E7 A7 E7 E7 A7 A7 E7 E7 B7 A7 E7 B7 The CHORD E7 is made up of the notes e, g#, b, d, so during a measure marked E7, you can use any note in the key-signature as a passing tone, but you want to park on one of the notes e,g#,b,d... or at least hit them on the beats to re-emphasize the current chord. The Key for a song in E-major has 4 sharps (circle of fifths), so the SCALE for E-major has e-f#-g#-a-b-c#-d#-e. The BLUE notes are the 3rd and the 7th of the scale. This means you want to hit the 3rd (g# in Emajor) flat in melodic lines. Vocalists and violinists can do this. Guitarist would bend a g up near a g#. Piano players can't do this, so they do a grace-note from g-g# to say "I really want to play the note between these two, but I can't." For the Blue 7th, don't bend it, just play the note a half-step down (flat). E-major wants the 7th to be d#, so in the Blues, play d-natural instead. When you get to a chord marked (IV7) A7 in this example, the chord A7 is the notes a-c#-e-g... in A7 measures, spend more time on these notes. The major scale for A is a-b-c#-d-e-f#-g#-a. Hit the 3rd (c#) flat (c-natural). (((The blue-third is ambiguously between the expected major-3rd and the half-step-lower minor-3rd... so sometimes it's one and sometimes it's the other, and sometimes it's a bent-note somewhere between the two.))) Because A7 is a 7th chord, and the Blues, replace the g# (the 7th) with a g-natural.
It can seem pretty daunting at first, but Music Theory is also a study of PATTERNS. It's a way to NOT have to learn everything by brute-force-memorization... by recognizing the patterns, you can predict the answer. And as you do it over and over, it becomes subconscious, so you stop thinking about it.
Remember that no one knows everything, so don't worry if you can't learn everything at once. Learn a little bit. Then learn a little bit more. It's a journey. If it feels like work, then you need a different job. Take a break from learning for some months. Keep playing music. Then learn a little bit more.
A simplistic answer. Think of some of the songs you know and play, in the same key. Likely as not they will have the same main chords. As in C, they'll have C F and G. This occurs in just about every song in C. So next time you start to learn a song that starts on C, you won't have to ask what the other two main chords are. You probably already 'know' a lot of theory - it's just that you don't know you know! Patterns like this become the theory. Look a lot deeper into what you play, and how it's put together, and look for patterns.
The only way the parts of music theory relate to each other is that they all relate to music practice. To the creation and performance of music. The parts of music theory were all created at various times in various places by various people, and they sometimes conflict with each other. There is an expression that would say “that is why it is called music theory.”
Music theory is useful for improving your knowledge of music so that your music practice improves. For example, if you write songs, and you only know one major scale and one minor scale, then discovering the Klezmer scale or Egyptian scale or Blues scale can be a revelation that leads to writing more interesting songs.
If you want to learn music theory without a teacher, then you need tutorial books and videos and lots of patience. But the thing is, you can go a little at a time. I suggest you learn a little music theory and then put it into practice and then learn a little more theory. If you don’t know what a Blues scale is, start there. Learn what it is, then learn how to play it. Then continue exploring.
There are many great answers here already, so I'll add my US $0.02 and keep it short.
"Theory follows practice". For centuries musicians have just written songs and pieces of music. They practiced the art of music; that came first. They didn't think much about theory. Then at various times academic scholars have come along and asked themselves "How did those great composers write those great songs? Are there patterns and methods that we can deduce by studying what they wrote? Will our understanding of those patterns help us to make more music?" And that is music theory, which comes after musical practice.
I often say that if you are a musical genius, you don't need much music theory. You just make music. I think music theory, on the other hand, is for those of us who are not musical geniuses. Us average musicians can learn useful things by studying the theory underlying how the geniuses go about writing songs and music. Thus we can improve our own music-making.
The quick bottom-line answer to your question is that chords are derived from a given scale in a particular key.
Ex. - Given the Major scale in the key of C, the derived chords are C, Dm, Em, F, G, Am, Bdim, C. These are referred to (in order) as the I (one) chord, the II (two) chord, etc ... III, IV, V, VI, and VII chord. When the major scale is harmonized, the I, IV, and V chords are Major chords, the II, III, and VI are minor, and the VII is diminished.
The key for you is to understand HOW the chords were derived/constructed. The short answer is "stacked thirds". Chord "quality" (major, minor, diminished, augmented) is determined by the 1-3-5 triad for each chord.
You may have to watch this 2 or 3 times, but this video explains it all very well: https://youtu.be/nAo7SyrTCmY?list=PLibUwyBuve0KkJD4FOTnHzzDQ-GVVafez