A brief note on plywood. Plywood is not the best sort of wood by any stretch of the imagination. It's convenient for framing large structures, and is very cheap, but is difficult to do any sort of detail work. It's composed of layers of offset wood pieces, and it splits when worked too close to the edge. It has unpredictable gaps, and just requires a slower pace. But it's readily available at at big box store.
This project uses a small amount of wood, in both 1/2" and ±1/8" thickness. You'll find that wood isn't sold in laser-precise thicknesses, so when making any selection, look for pieces that are closest to the measurements you need.
Look for pieces that are as straight as possible and have minimal gaps or cracking on the edges. If you get hardwood boards, the sort with a hardwood veneer on the surface, try to find pieces with as little damage to the veneer as possible.