U got it Scott, singletrack is the tight and (hopefully) twisty stuff. Doubletrack and fireroads are more open and generally made by ATV's and four wheelers.
As for a bike under 2K? You'll get a ton of different replies on that one. It really depends on tons of factors and there's tons of options within that realm. First rule, no department store bikes, LBS (local bike shop) only. LBS backs up their bikes, makes adjustments and has experienced mechs. assemble their rides. Then, it comes down to what type of riding you plan to do, XC, freeride, downhill etc. Also, your size and any health ailments (bad back and such). Keep in mind, most bicycles aren't made for 200 pound guys, bigger guys gotta pay for a solid bike to keep from constantly breaking/ wearing out parts. Then u gotta decide whether to go rigid, front suspension or dual suspension, all of which u can get in that price realm. Also, u don't wanna unload 2K on a bike you're not gonna ride, so being you're first bike you may not wanna dive that deep into the wallet. There's pro'ly tons more to consider too, but that covers a good little base there and my recommendation is to make some trips to the bike shop, test ride ecverything in your price realm, then goto another shop with different brands of bikes and test ride all theirs. Have an idea of the type of riding u wanna do and let the shop personnel steer you in a particular direction. Also, read some mtb mag back issues and get their bike reviews, those guys ride tons of bikes a year and can distinguish good handling and performance characteristics in a bike far better than the average schmo who rides only a handful of bikes in a lifetime. Also know that u gotta decide what size wheel bike you're gonna like in this whole schpeal, 26 inch bikes were the standard, but there's tons of guys riding 29ers right now, so test ride both. You've got your work cut out for ya man, but the main thing is to just get out there and ride, don't ride over your head and enjoy the sport, also, ride on legal trails only, nothin sets the sport back further than guys riding illegally. Good luck. I'm posting this from my phone, so if I missed stuff, or peeps already gave good advice, totally ignore me.