I want to fasten 1/2 " - 3/4" boards onto 2x2s. I'd like to scew them from the back so that they cannot be seen. What size screw should I use?
If you go straight through the 2x2 into the thin board, you'll only have a few threads in the board and it won't be very strong. If you use a good wood glue as well, it will be much stronger.
This isn't something people pay much attention to in most cases, but a real wood screw is tapered - the shaft isn't uniform in diameter. Likewise there are tapered bits meant to drill pilot holes for wood screws. This will give you a little more strength. Tapered screws aren't hard to find, the tapered bit may be.
An alternative to consider - a pocket hole set. The Kreg brand is most popular. They sell a special drill bit, jig, and screws that work together to make a stronger joint.
I believe they had a starter kit available at the big box stores that wasn't too expensive.
2x2's are actually 1.5" x 1.5", so the screw should be shorter than that distance plus the thickness of board you are using (1/2" or 3/4"), taking into account how far countersunk the screw is.
If you use a flat head screw, it will countersink, where as a pan or round-headed screw won't countersink (at least, depending how hard you drive it in).
The biggest issue is that the end of a wood screw is relatively narrow, so you're not going to have a great grip on the 1/2" board. If it's not supporting any weight, then you can probably get away with some #8 or maybe even #6 screws. However, if there is extra weight that will be on this board, then a #10 or larger will have bigger threads that will allow a much better grip.
Either way, you should drill holes for the screws all the way through the 2x2 to avoid splitting it -- these should be slightly smaller or the same size as the screw shaft.
If you're close to the edge of the 1/2" board (or it's a narrow board), it's a hardwood (as opposed to soft wood) or using #10 or larger screw, I'd also drill a pilot hole (slightly smaller than the shaft diameter) in that. Obviously, be very careful not to go all the way through (a piece of tape around the drill bit makes a great depth guide), and the best way to keep aligned is to hold the board in place and put the drill bit through the holes in the 2x2.