Just how much woodwork do you plan to do? If you're just going to build a pair or two, you might be better off renting tools. Home Depot rents and you should be able to find independent hardware stores that rent, too.
You can also have a woodworking shop or cabinetmaker make the cuts for you. It will cost less than you expect. The biggest expense is finishing woodwork. It's labor intensive and you'll get charged for it. The good news is that you can DIY the finish with a minimum of tools.
If you want to build, you'll need a tablesaw and a router at minimum. Those will exceed your budget if you get halfway decent ones. A jigsaw and a power sander are nice, too. You'll also need a drill and a number of clamps.
My advice would be to find a woodworking shop or cabinetmaker to make the cuts. They'll charge something like $1-$3 a cut or maybe $60-$80 an hour. But if they charge $80 an hour and spend 30 minutes cutting your MDF, $40 is very reasonable. You'llprobably spend about the same renting tools, but a cabinetmaker will have better tools and will make beautiful cuts. You can take it from there with your own drill, sander and finishing.