I am making a control panel out of wood, and I need to cut out a small 3" x 3" square hole in 1/2" plywood. I only have a small handsaw, with a wide blade. (about 2" wide) I don't feel comfortable using power tools for cutting, (*Jigsaws, reciprocating saws, band saws, etc.) so that is not an option. If I even cut this with my saw, how should I do it?
You're not going to be able to reasonably cut the square out with your hand saw. You'd have to scrape through the wood instead of cutting it.
Pick up or rent a coping saw and a hand drill (See saw below). Drill a hole in the four corners, remove the blade from the saw and run it through a hole, hook it back up and saw away.
Let me suggest a "Japanese" style pull saw. These are available in many sizes and are able to do both plunge cuts into your workpiece as well as cut right into a square corner.
Something like this:
These are on the high-end but you can find similar saws at most home centers and hardware stores. They are very useful!