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.  

https://www.kregtool.com/store/c1/joining/

I believe they had a starter kit available at the big box stores that wasn't too expensive.