If I understand your question correctly, you'd like to get a hands-on practical understanding of "weird" chord changes that seem different from simple things like C - F - G - C.
Very often, people asking these questions are bombarded with theoretic terms and concepts, but I'd like to try a different approach. My alternative idea is to have you play things and get to know the chords and their roles in each "story" first-hand. Touch them, feel them, play with them, see how they interact with each other. A bit like you get to know people - through practical interactions. Practice first, not theory first. In music, your feelings are always "correct", even if you don't know how to dress it up in fancy legitimate mumbo jumbo.
Try playing the chords: C - F - G - C. Then play the same, but replace the G with Ab/Bb, i.e. right hand playing Ab major chord and left hand playing Bb as bass note. What happened? Weird chord, those notes are not in the key of C major? :) Then play it once again, but hold that Ab/Bb and try what it feels like if you play Eb instead of the final C. That Eb suddenly became a "base chord" or "home chord", i.e. the harmony feels like it it resolved and came to a conclusion. Do you agree? Can you feel it? But you can also play C instead of Eb, and then that is a home chord. So, when you have that Ab/Bb chord there, it opens up a possibility to go either way, either to Eb or to C, and whatever you choose, it will feel like the home chord. ("tonic" as they say in their fancy mumbo jumbo)
But I can sympathize with wanting legitimate logical explanations. Think of it this way: musical keys (as in, the key of C, with the major scale and chords C, F, G, ...) are much, much less rigid and much less monolithic than you might have thought. By introducing "strange" notes, you change the "harmonic context", and open up new possibilities. It's like wax or clay, you can mold it in many ways, and there are always many alternative paths the "harmonic context" can take. If you're in the key of C, but you play a C7 chord which has the Bb note, it's like pretending for a bit that you're slightly in the key of F... So, when you encounter a strange chord, and cannot see a "I-IV-V" of that key, you might be able to see a "I-IV-V" (or I-II-V-I or whatever you are familiar with) sort of pattern of a different key superimposed on top of your original key. Both keys exist in that moment in some way, as possible alternative ways to go. Some people say that you "borrow" from another key, or you visit that key. The longer you keep utilizing the notes and chords of that other key, the more you establish that as the prevailing key. After a few bars, it will start to feel like you're not visiting the neighbor's house anymore, you actually moved in. ;) (If the change is really permanent or at least semi long term, this is usually reflected in musical notation with a key-change, i.e. a new key signature marking. If the visit is only relatively short, only temporary accidentals are used for those notes that happen to step on the modified pitches. Quite often, none of the written-out melody or other notes happen to hit the places where the harmonic context has temporarily changed - and then you just have to know it by understanding the chords. If you are improvising a solo, you need to be aware of what happens in the harmony.)
To give another example. If you're in the key of C, and you go like C - F - G - C. Try C - F - D7 - G - C. When you have that D7 playing, it's like pretending for a short while that you might actually be a bit like in the key of G major, where the basic chords are G, C and D (or D7). Agree? Many interpretations! In my subjective experience, I feel that there are only relatively few basic underlying mechanisms and "tricks", and even jazz harmony uses the same basic elements and tricks, just a heavier dosage of tricks. Theory guys will want to explain things using lots and lots of names and concepts, but unless you play with the things yourself in practice, the names and concepts will only distract you. You should spend 10 hours playing for every 1 hour of theory or Youtube videos. :) And by playing I mean like child's play. Chords should be your toys, play with them.
Edit: that might not really explain a "method" yet. What I'm trying to say is, find out and get a feeling of the roles the chords have, and if and how the roles seem to change when you encounter a "weird" chord. First, simple songs with simple roles, then advance to more complicated things. Basically, the reason you don't understand a chord progression is, you haven't played enough with simple chord progressions. Seemingly "complicated" or non-obvious progressions consist of the same basic elements as simple and obvious progressions, just more of them. In order to understand what "great-grandmother" or "grandmother" mean, you have to understand what "mother" means. And to really understand that, you have to know a mother, or be a mother, or something. If you don't know first-hand what the role of "mother" means in practice, how much do you think a biologist or psychologist's explanations are going to help, even if they got a diploma and know words you never heard before?
If you don't use the chord roles, you don't really understand them. If you take a ready-made chord chart that someone else has written out for you, you're not really using any chord progressions or tricks. They made all the decisions and only gave you the end result. But you want to know what happens in the decision-making! In order to understand the roles, you have to know what would happen if there was something else instead - and to know that, you have to try it yourself.
Learn to play simple songs by ear and find chords to the melodies yourself without looking at ready-made chords. This is absolutely essential. You have to develop a sense for basic chord roles, I - IV - V. Start with songs like London Bridge is Falling Down, Incy Wincy Spider, or Happy Birthday, and only use the most basic chords at first. Do the chord-finding for as many melodies as you possibly can, and then some. I have to repeat, practicing chord-finding is absolutely essential. There is no other way to "get it". Practice, practice, practice. You only learn it by doing it. Don't think for a second that there could be a way to sense chord roles through reading or other kinds of theoretical studying. Reading text, no matter what text, just does not produce musical feelings. (unless you're a fantastically good music reader who hears the sounds by looking at written music) Trying to learn chords theory-first is as absurd as suggesting that it would be possible to develop a taste for food and cooking through textual descriptions of food ingredients. :)
Do the chord-finding exercise in multiple keys, i.e. raise or lower the whole song to a different pitch, so you see the relative roles. Feel how different chord choices affect your feelings. Apply substitution tricks to the simple songs. Learn another song that has a different chord progression. When you see a weird chord, think at each point of the progression, where do you feel the tonic i.e. home-base chord is? Is that the only alternative?
When you play a seventh chord, can you feel what chord it's trying to imply should come next? Take a minor chord progression: Am - Dm - E7 - Am. Try inserting an A7 in front of the Dm, so: Am - A7 - Dm - E7 - Am. Can you feel how it made the urge to go to Dm stronger? Then try Am - Dm - B7 - E - E7 - Am. Doesn't the B7 very strongly lead you to E ... and when you add the seventh, it even more strongly than just plain E, leads back to Am? (that could be called the V - I motion...)
When you learn any such chord trick or mechanism, try applying it to songs and chord progressions you know. In a song that goes in C major, try substituting an F major with a D minor. How does that feel? Try placing major sevenths as in a V - I motion, in different places. In a song that's in C major, if there's a C - F progression, try doing Gm - C7 - F. Can you feel how strong the G minor is there? Is it even stronger than just simple C7 - F? And how nice it is when you go G major - C again. How about making it a Gm6? Then move the bass to C, so Gm6/C ... (in fancy talk that's also called a C9 chord by the way)
Mess around with the chords. Don't accept ready-made decisions as the only "correct" one. Try applying any trick you know. When you encounter a seemingly weird chord that still works, try the same trick somewhere else. Play with the chords, they are your toys. There is no "correct" way to play with toys, you know.
Listen to music from records, find out the chords, and try different chords instead. Find out what tricks other people used. No amount of theory is going to make you understand the role of different chord changes, you have to try it yourself. Theory can only provide ideas to try, and help with reasoning about things with other people.