Developing The Left Hand Chords

I. Finger Independence:

Ok so the first thing I learned as a guitarist was finger independence. You have 5 fingers, of which 4 are on the fretboard (well, there are some exclusions but let's not get into such details, you'll rarely use the thumb) and they form chords or play licks, do scale runs, mute, etc etc. The aim of this part of my lesson will be to cover some practice patterns which will help you develop your fingers as if they're your kin, not just part of the hand.
1. Chords:
A very important part of your playing is the use of chords. You won't be able to play good music if you only play power chords. Well you might be able to but not everybody is Dan Donegan (Disturbed's guitarist) or Sully Erna (Godsmack's rhythm guitarist + singer). Even they use different kinds of chords (Sully Erna's solo album is filled with examples of how a guitarist and vocalist who's used mostly power chords can easily outmaneuver his own skills + display quite the knowledge in playing).
Soooo... let's start with something simple
Dmajor to Amajor for example:

E-2-0
B-3-2
G-2-2
D-0-2
A-x-0
E-x-x

Looks simple enough, right? Try playing it, you'll notice how your fingers are moving in each other's way, right? The aim of this simple exercise is to learn to play it SLOWLY I repeat SLOWLY, at say 20-30 bpm, when you master it at this speed increase a bit until you can do it by heart. The aim of playing this basic progression is to force each of your own fingers to go towards the place it has to be. You'll see soon how at 20 bpm your fingers no longer push each other as if they're bowling pins, but dance on the fretboard. Now let's move onto something harder
A aug to E major:

E-x-0
B-1-0
G-1-1
D-2-2
A-3-2
E-4-0

Now this is harder due to the fact that you're using all your fingers on the A augmented chord and all of a sudden you end up using only 3 of your fingers, not to mention going back to the A augmented. However when you practice this at low speeds you'll be able to take off your fingers after you play the chord. The main idea of this exercise is when you play the chord, take off your fingers and place them properly on the other chord and then back. I intentionally give examples with 2 chords because that way you constantly have to switch from a wider stretch to an easier one and back. When you get the hang of it, let the chord ring out and with the second beat jump to the E maj chord, let it ring and following the next beat, return to the A aug. Rinse, repeat. Easy, yep :) That's why I'm labeling this lesson For Beginners after all.
2. Arpeggios:
Now let's get closer to the classical music. You've heard of notes ringing out sounding perfect together like in, for example, Moonlight Sonata by Beethoven. If you add the notes together, you'll have a chord. Arpeggio means broken chord (Italian, like most of the musical definitions) and it is exactly what it means - you break the chord and play each note individually letting it ring out. For example we take a D Major chord

E-2
B-3
G-2
D-0
A-x
E-x

Now we play it in an arpeggio:

E----2---
B---3-3--
G--2---2-
D-0-----0
A--------
E--------

Easy enough, right? :) The picking pattern is simple, you just go down and up, the left hand keeps pressing down the frets of the chord and does not let go. Now let's make it harder (not really):

This is a D major to D minor change. Did your fingers mess up? Of course they did (if not, skip to the next part) because the way you finger the D major is different from the way you finger a D minor (if not, please record yourself playing Vivaldi's seasons and send it to Guiness World Records, I'm pretty sure they'd count it as a world record, you prodigy!) Now you notice how you have to move your fingers a bit more, don't you? But not only do you have to change their position, you've got to time it with your right hand. Tough one, ain't it? Don't worry tho, this is taught in elementary school haha. Practice it slowly once more, until you reach perfection, rise the speed a bit after 10 successful goes and admire your work.
Now onto the harder change:
E dim to G maj7:

Here you'll experience some messing ups because you not only use all the strings when the chords are formed with 4 strings but you'll also have to move your hand a bit to the right and then back to the left. However it's simple as pie, all you've got to do is when you play the A# note on the E string you move your hand so that your index finger is at the second fret, you fret the high E string's F# note with the index whilst stretching the pinky to the fifth fret of the D string. Now goes the simple part how you glide your right hand down and then up the chord just to return to the simple E diminished chord. Easy as making yourself a sandwich :)
Now onto the last exercise in this part of my lesson:
Arpeggios (once more)
I'll show you how you can spice up your power chords in this part, featuring a little stretch + lots of fun for the guitarist :)
Here we have an E power chord

E---
B---
G---
D-2-
A-2-
E-0-

to play it in an arpeggio would mean to play it like this

E-------
B-------
G-------
D---2---
A--2-2--
E-0---0-

now what we do is we move the middle finger to a new position and we get the following bastard to play

E-------
B-------
G-------
D---3---
A--2-2--
E-0---0-

Now combine both the "triads" we made. It'll sound nice and it'll also be quite easy + it will develop your middle finger's strength. Now that you've got the basics of this exercise try moving the index to the third fret instead of the middle to the third fret:

Easy as 2+2 ain't it? Now use a power chord which features all the fingers and try the same. Now this ain't a lesson as in "finger the fifth fret with your index, finger the lower string's 7th fret with the middle finger etc", it's supposed that you can do that already, if not learn that and then come back. When you've learned to deal with such small shifts try to play them with a hammer-on (guess what, you'll be adding the pinky to the play). Now try to do the same with a different chord (not necessarily a power chord). Now do it with a hammer-on. Sounds hard but when you practice it SLOWLY you'll reach the point where chords are easy for you and your left hand is slowly beginning to advance in the ladder of finger independence :)
This concludes part 1 of my Left Hand Development lesson, hope that you find it useful and that you cope with my decision to label it beginner, it really ain't that hard and I've tabbed most of the stuff I do :) Besides, I learned to play this stuff in my third month of guitar playing and you gotta admit, a three-month-old guitarist IS a beginner. Constructive criticism is more than welcome, to be fair all sorts of criticism is welcome :) Don't hesitate to comment or ask if something ain't easy for you to understand, I only wish to help and share my experience :)

In that last part, I don't understand how you can move your index finger from the 2nd to 3rd fret (A string) while your middle finger is still on the 2nd fret (D string). It's just impossible to cross the index over the middle like that. I can do it if I move both fingers to new positions but not like how you mentioned.

This is a really great post especially when a lot of people can forget the fundamentals when they go for the harder stuff (myself included sometimes). Helped me a lot
But yeah, I also agree with dabutz about the last part. Its nearly impossible to move your index like that...