What Should A New Mountain Biker Buy?

It'd depend a bit on what type of riding you intend to do. If you're getting into Downhilling and jumps you should get some pretty serious protection. Full-face helmet, knee/elbow pads, spine protection, etc.

If you're settling for cross-country, a helmet and some gloves will go a long way.

Apart from that there's cycling shoes and SPD-type pedals, paddded cycling shorts, either tight or baggy. A sweat-friendly jersey. A book on bicycle repair. Even if you have a shop do the work it's good to know what it's about. Some basic tools, every rider should be able to fix a flat.