I know each set of control arms does a given job. What does the Front Uppers do? Front lowers? Rear uppers, and Rear lowers?

I know one fronts push axle forward but don't know about the other set.
And same with the rear.

Lets say I wanted to push my entire axle forward 2", what would I need to get? I really like the stance of the H1.

 

Uppers and lowers do the same thing, just backwards from each other. If you want to move the axle forward/backward in the wheel well, you need both sets.

Front upper -
Lengthen to lower caster / raise pinion
Shorten to raise caster / lower pinion

Front lower -
Shorten to lower caster / raise pinion
Lengthen to raise caster / lower pinion

Depending on lift height, uppers may not adjust short enough to get the caster where you want it. So if you only get one set, lowers are the better option. And in the rear, uppers are the better option.

Getting both sets gives you more adjustability, which gets more important with taller lifts, especially on 2doors where you want to move the axle also.