I've been on a run to get a lot of theory down after realizing how amazing music theory is. As a guitar player going through a intermediate-advanced theory journey, I would like to know where 'Chord-inversion' learning comes in. Is it something that is like a collateral learning? Or does it require an allocated plan/time to get it all down? And what are the applications?
Knowing chord inversions is very important because it will allow you to play any chord in almost any position. This will increase (left-hand) efficiency while playing chord changes, and, more importantly, it will make your changes sound more smoothly due to the more natural voice leading.
You could combine learning chords and their inversions with learning scales and modes. Start with triads and play all triads of a given scale/mode on all possible sets of 3 adjacent strings up and down the neck. You should try to realize that you can see the chords as subsets of the scale patterns, so combining learning chords and scales should feel very natural.
E.g., if you're practicing the C major scale and if you happen to choose the strings d-g-b for finding all possible triads, you would play
1 3 5 6 8 10 12 13 0 2 4 5 7 9 10 12 2 3 5 7 9 10 12 14
which corresponds to playing all triads of C major in their first inversion. You should also do this with the root position (starting with F in this case)
1 3 2 4 ... 3 5
and with the second inversion (starting with Am)
1 3 2 4 ... 2 3
Then try to see how these shapes change when played on other combinations of 3 adjacent strings. Finally, find all open voicings by using non-adjacent strings, such as e.g.
g 5 d 2 A X E 3
for a C major triad.
After having figured out all possible triads in this way (and if you're still hungry for more), you could do the same for seventh chords. Here you obviously need combinations of 4 strings. Adjacent strings will give you drop-2 voicings (or the few playable close voicings), and other choices will give you either drop-3 or drop-2-4 voicings.
Inversions are pretty easy once you understand what is happening conceptually. The majority of your time will be spent applying what you have learned to the guitar.
Contrary to other answers, I would say application for guitar is very important. Inversions will help you identify more comfortable chord voicings. They will also help you create smoother voice-leading between chords in a progression. This is important for soloing, jazz comping, fingerstyle, or if you are a singer-songwriter.
They are very important indeed, particularly for playing chord melodies in jazz, contemporary, and any song where you want to combine the melody and accompaniment. It benefits a player in so many ways to learn four different inversions for maj7, min7, and dom7 chords on four-string groups. To clarify, here's an example of maj7 chords in root, 1st, 2nd, and 3rd inversions (not nec. in that order in the diagram) on string 1234 and 2345:
You can find more diagrams like this on the web, or buy a book that has them.
Learn these! Some are easier than others, but learn all of them, because you can use all of them to alter chords (such as b5 or #4, 9ths, etc.), and some of the hard ones are easy to play when altered as such, and thus very useful and beautiful sounding.
A good player can use these to take many songs and play them with both melody and accompaniment solo style. Also, learning inversions will help you become better acquainted with the fretboard, open your ears to new sounds, and make you a better-rounded musician.
It also helps in that it makes you think of chords in non stereotypical ways. For a great many guitar players power chords, bar chords and open chords are the only ones in existence but it does not need to be like this.
Too often chords are just thought of as background noise to full the air while some sort of lead playing goes on but it needs not be like this. You can easily use the full breadth of chords to give shape and substance to your musical endeavors.
Chord inversions don't come in all that much in guitar playing from a theoretic level.
For the guitar, you usually have several chord shapings for what is named the same chord. If some note is shared by two chords in sequence, you tend to play it on the same string. If you do chord changes mainly staying in the same position (if necessary, by judicious application of a capo), you'll tend to pick sequences of inversions that have reasonable voice leading.
Contrast this with chord play using barré chords where you basically use the same shape while shifting positions. That does not really involve noticeable voice leading unless very consciously changing rhythmic picking patterns to keep related notes in a related rhythmic position (even if they are played on a different string).
The usual use of "chord inversions" is for picking out the bass notes to strike in a picking pattern while the melody notes more often than not tend to be whatever shape happens to occur in the upper three notes.
It is just for the final chord/picking pattern that you usually want to end in a position where the last note again corresponds to the bass note of the non-inverted chord.
So in a nutshell: there are applications, but they are very much reined in and constrained by the realities of the guitar and its string layout and playing style. The limited degrees of freedom as compared to a keyboard instrument give much less leeway to give theoretical frameworks a nice workout: practical considerations dominate the choices of inversions.
Basically, you get to pick out bass note and alternate bass note for patterned accompaniment. That's actually what happens with accordions as well, and accordions indeed have two button rows for basses, four button rows for chords, and the assemble the chords from the same basic octave of notes, never mind inversions as you cannot change chord shapes anyway.
The guitar, in contrast, has at least a bit of leeway for touching up progressions in the three to four chord strings that don't make musical sense. But overall, the principle "get the bass notes right and put any old inversion on top" holds pretty much for guitar play as well.