Pre drilling the holes both helps prevent binding and makes for a tighter joint.
When doing this use the screws you'll be using as a guide to choose the proper bit. I hold the bit up to the screw and judge the diameter. What I like is a bit that's basically the same size or just slightly smaller than the shaft of the screw, not counting the threads. That way the threads still have wood to bite into to hold.
The more dense the wood the more important it is to pre drill.
With the 2x2s, you have to consider the usual use for those boards. They are rarely used for anything cosmetically important in normal construction, they usually just serve as a piece to nail to.
Due to their narrow size, they are much more prone to warping as they dry out after cutting.
Taking any 2 by board, 2x4, 2x6 etc, and ripping it into 2x2s always gives a much more usable board. Just choose good straight 2x4s to begin with.
You can probably get a table saw suitable to do the occasional project for $100 or so.
The way I rip my 2x2s is I use the edge of the board to set my guide, then lay the board flat to rip it. Be
extremely careful doing this though, a mistake can cost a finger, because they'll be in relatively close proximity to the blade.
The reason I use the edge of the board to set the guide is, as you have probably noticed, a 2x2 isn't 2 inches by 2 inches, it's 1.5 x 1.5. Using the edge of the board gives you square pieces to work with.
When ripping a 2x4 you end up with two 2x2s and a this 1/4" piece left over. I use those pieces as backstops on rat racks to prevent the tubs from being pushed in too far.
Here's a couple of links to pages on how I build my rat racks.
http://www.arbreptiles.com/cages/rat_rack/ratrack.shtml
http://www.arbreptiles.com/cages/rat_rack2/birthing_rack.shtml