Which one are you getting? If you are getting a cutter in the GO! line, then you are limited to GO dies, and as mentioned, Accuquilt controls the price. If you are an ebay user, you can find used dies. You can't use sizzix dies as they are too large. If you are getting a Studio cutter, then you have more options. You can use studio dies, or any other commercial 1" thick die. With a shim, you can use sizzix and/or go dies.
As far as which dies are most useful, only you can answer that. What types of quilts do you make? Do you cut a lot of strips? Squares? Triangles? You might want to pick a favorite pattern, or choose one from the accuquilt website and buy the dies for that pattern.
The mats are sized for the dies, and in general, you can use a larger mat with a smaller die, but if you used a mat smaller than the die, you would only cut part of the shape.
The accuquilt site has a video for almost every die. There are also tons of youtube videos.