The little inner levers move from big-to-little (the derailleurs are spring-loaded so don't need much force, just a tap when going with the spring).
The brake levers move from little-to-big (it's longer than the little inner lever so easier to move against the springs).
The left hand controls the front derailleur. Pressing the left inner lever will shift you onto the smaller chainwheel and make pedalling easier but progress slower. Pressing the left brake lever will take you back to the big ring, so pedalling harder and going faster.
The right hand controls the rear derailleur and the effects are reversed.
Pressing the right inner lever moves you up to a higher gear (harder work, faster progress)
Pressing the right gear lever moves you down to a lower gear (easier pedalling, slower progress)
Cross-chaining, as mentioned above is when you have the chain running between opposite extremes (big-big or small-small). Folks get wound up by it, and it can cause the chain to rub on the cage of the front derailleur and make a rattling noise and wear components our prematurely, but it's not the end of the world for short periods.
Think of the front chainrings as 'high and low range' and the block of gears at the back (the cassette) as the finer selection. The gear ranges provided by your front chainrings will overlap a lot, so try to change chainrings (left hand, remember) before you reach the extremes of the cassette. If you drop to the inner ring at the front (inner click, left hand) preparing for an upcoming hill for example, then you'll probably need to click the inner right lever too to shift you back up a couple of sprockets at the back. This smoothes out the transition between the higher and lower ranges. And vice-versa when shifting back to the big ring - use both brake levers to shift up to the higher range but also up a sprocket or two at the back to keep your pedalling effort consistent.
You'll definitely use your right hand more than your left - that's natural.
