The Big Shot is a great machine that will take the steel rule dies - which are the thick dies you'll need to cut a few sheets of cardstock at once. As for which dies to get, it all depends on your style. There are soooooooooo many to choose from!
If you're applying the paper to wood, I would try Mod Podge as your adhesive. Paint a layer on the paper, press it on the wood, smooth out any bubbles, and paint a layer over it to seal it.
It comes in all kinds of finishes (matte, glossy, glitter etc).
Good luck with your project and welcome to the neighborhood