After step 1, we have a solved part of the cube, and we have three sides that we can move freely, without breaking what we have accomplished. Not bad!
In step 2 we expand what we have to a 2x2x3 block. That is, we add one corner and two edges to the solved block, going from 4 to 7 solved pieces of 20.
This is quite similar to how you work in step 1, join the corner with one of the edges and work from there. When you're familiar with step 1 you should have no problem.
But be sure not to break up the 2x2x2 block! You need to be aware where it is at all times. Otherwise it's back to step 1...
The hardest positions are when a piece is in it's right position, but twisted the wrong way. The easiest way to deal with that is to just pick another corner to solve!
These simple and very useful moves (to the right) took me months to figure out (actually I didn't, someone I taught showed me...), since it involves breaking the holy 2x2x2 block. As an exercise, try to find how to do the the third example cube above in 7 moves instead of 9 by breaking the 2x2x2. It's not easy, so here is a hint: First move: turn green layer 1/4 clock wise. Second move breaks the 2x2x2 (highlight to show hint)
Sometimes, the first move of step 3 will be the opposite of the last
move in this phase. So with some foresight you can skip both. Also, while doing the last turn of
step 2, you can often stop half way, and fix a pair of bad edges midway. See the examples page
for several examples of this.
For more advanced techniques, check out the block building page.