Whenever possible, use bits with 1/2"-dia. They provide better stability with less vibration, and they typically produce a smoother cut and have longer cutter life. Except for very small and very large profiles, router bits typically are available in both shank diameters.

Notice that many router bits are available in both a 1/4" and 1/2" shank size. Whenever possible it's best to choose the larger shank size. The 1/2" bit shanks are substantially stronger and less prone to breakage than a smaller shank. In fact, a 1/2" shank has nearly four times the surface area of a 1/4" shank.

Yes, you can use a router on plywood.

These bits cut right into the material to create a groove or dado, a groove across the wood grain. They are also typically used to hollow out areas for a mortise, essentially a hole, groove, or slot into which another piece of material fits.