You should probably start in Short Course 2wd, for a number of reasons...

1) It's very popular (I'd be surprised if it isn't the most popular class at the track)
2) The vehicles, parts and electrics for this size and scale are relatively cheap and durable
3) Short Course trucks have closed wheels (body covers tire treads), which makes racing easier, you can bump people and not ruin theirs or your own race, whereas in buggy or truggy racing, wheel contact usually turns one or the other over. Rubbin's racing, boi.
4) They look really cool
5) They're relatively easy to drive.

There are MANY good 2wd short course trucks.

Since you're venturing into racing, I strongly suggest you get a kit and separate electronics. You will learn a great deal just from building the kit and fitting the electronics. In order to achieve this you will need...

The car itself, in kit form or as a 'rolling chassis' (rolling chassis usually means 80% prebuilt, a lot of cars come like this, but I still suggest you build your own). The mainstream cars in this area are as follows, in order of quality, best first...

Kyosho Ultima SC (may be a little on the expensive side for a beginner, but a good investment nonetheless).
Team Losi 22SCT
Team Associated SC10 Factory Team
HPI Blitz ESE Pro

Any of these cars above, properly setup and driven, would be capable of winning a major national race.

The tools required in the manuals of these cars, but basically that means: Screwdrivers, hex drivers (aka Allen keys), a body reamer, paint for the body, threadlock (blue), tire glue (any good cyanoacrylate) and possibly a few nut drivers (though box wrenches supplied with all kits cover you for most needs).

For electrics for any of the above, you will need...

A speed controller and motor combo of the '540' variety (this refers to the length of the motor can). Talk to the hobby shop or guys at the track about this, as it may be worth you buying whatever they insist on their stock racers using, for example a specific motor that must be used according to track rules, to make sure that everyone is one a level playing field.
A few 2S lithium polymer batteries (suggest 2-4 depending on budget to start with. There are many ways to get cheap lipo batteries (www.hobbyking.com for example), NEVER buy these locally, you will get badly ripped off.
A steering servo (would recommend something with 100 oz-in of torque or more and as much speed as you can afford)
A good charger (extra spent on a good charger is never a waste of money in my opinion, get the best you can afford)
A radio set. No need to think about this one, a great radio is available for the beginner racer and its only 40 bucks. The FlySky GT3B or GT3C.

Final note...many tracks run a Slash spec class. This means that everyone is using a 2wd Traxxas Slash (same kind of truck as these above), this can be a really fun class to race in, but the Slash is not even close to a match for the cars mentioned above which are thoroughbred race cars.

Here's a taste of 2wd SC racing. These are good drivers on a technical track of course...

[youtube]http://www.youtube.com/watch?v=SdeQDr0r3vo[/youtube]