With a pack of the girds, they are sold by the number of cubes they will make. A 3 cube set is 13 grids, 4 cubes can be 16 or 17 grids, 6 cubes should be around 23 grids. Around here, a 3 cube set is around $20 and the one 6 cube set I have seem was like $40 (over priced though).
Look for sales, I have gotten some for $10 a set. Check online to compare prices, I usually search 'Wire storage cube' and the should bring them up. Also look for used ones, try kijiji, Craigslist or other sites. You can usually find them for around $5-10 for a set. I would also check garage sales, but those are usually in the summer. Thrift and second and stores might have them as well.
Try to figure out how many would would need for the cage you want to build so you can hopefully get enough. Think of the minimum (walls, roof), then add ones for levels, this way you can build the cage and build a level if you have extra or do it later if you get more.
I am not sure about what wood to use. Are you using it for the base? Most people seem to use stick on tiles to cover the wood. As long as you cover the whole thing including the sides, it should be safe to use regular wood.
