The above comments and advice work very well. My feeling for mounting pictures or what ever you want to cut onto plywood should be dry mounted. I have found this works best and there is no waiting around for things to dry. The only problem with dry mounting is the cost of the equipment. The next best method for me is what Travis does. Using spray adhesive. You can cut your puzzle almost immediately after mounting also. The down side is you will get spray glue all over the place.
I use 1/4 baltic birch plywood (5ply). Please dont use the cheap 3 ply for this. You will not like the results. Other puzzle makers will use oak, cherry, and poplar. I hear Poplar is a really great wood for puzzles but I have never use it just yet.
Stand up puzzles can be cut out of hard wood or plywood. Use at least 3/4" thick stock. I dont recommend using a soft wood like pine. Pine seems to break to easy. I use mostly sassafras and some oak for my Stand-up puzzles. The plywood works out great because of the strength of the wood. It will not break as easily as regular wood will. However I think plywood is rather ugly compared to hardwood. I use a fd#5ur blade for these puzzles. This cuts the wood really good and there is not much sanding needed after cutting. You can also use a #5sr but there is more sanding needed.