I've built a few cages & flights for my birds. For finches, you won't have a problem with a wood frame and you could also have the sides & back covered in wood - or you could use plexiglass. I like to leave the top wire and the front wire.
I use hardware cloth for the wire on my cages. With finches you won't have a problem, since they aren't apt to chew on it, but be aware that hardware cloth has zinc in the welds. My cages are certainly not show cages, but they serve the purpose.
You could also build the frame & buy a premade cage front. These come in various sizes.
I believe I have a few good links listed in the
Cages, Toys & Supplies subject area where you may find some ideas & supplies.