Rabbets. I made a quick, super simple tray today with 1/2" plywood sides and a 1/4" plywood bottom. The sides are rabbeted together (1/4" square) and the bottom is let into the bottom with a 1/4" rabbet. It's a little finicky, you have to go slow at the router table when rabbeting such thin plywood or you get tearout. (My particular project was a 24" square tray, so I was trying to use the lightest materials I could to keep the weight down.)
For a step up (and to use solid wood), the material thicknesses above have to beef up a little bit. At least the bottom would need to be 1/2". By then, you might also want to use a little better corner and bottom detail than a simple rabbet--move up to grooves and dados. There's nothing wrong with solid wood sides and 1/4" plywood bottoms, the engineered material will greatly reduce issues of wood movement.
Last step up is to start fastening corners with more showy joints... box joints, dowels and dovetails. You can still use plywood bottoms to avoid movement issues, perhaps 3/8" or 1/2" for large drawers.
I'm curious--does anyone knows any good rules of thumb for the lightest material sizes reasonable for small, medium, and large drawers?