Pretty much any dSLR will get you seriously nice quality pictures these days. Entry level Nikons or Canons are both great. For some reason, you see pretty much all of the professional photographers using Canon SLRs, but at on a casual level, a Nikon will do just as well.
If you are photographing specifically sports, you would want to get a camera body with a decent ISO (at least 1600) with not too much image noise. Don’t worry too much about resolution. Anything above 10 megapixels gets pretty excessive. For photographing sports, I wouldn’t suggest a full frame camera.
I would suggest investing more in a fast lens. A good lens will make a huge difference sometimes. Find one with a low f-stop of around f/1.8 or f/2. Those will get you sharp, crisp pictures. For taking pictures of racing, I would suggest a longer lens, probably at least 100 mm focal length. Unfortunately, long lenses with low f-stops get expensive, especially zoom lenses, but fixed lenses are a little cheaper. Zoom lenses of similar price will be a little slower, but will do perfectly well in nice sunlight but will struggle in dimmer light.
Finally, the most important aspect of taking pictures is thought. I think that the more thought you put into a picture, the better it will turn out. Think about your composition, think about your subject. Move around and look at it from different angles and perspectives. Putting in some thought will compensate most shortcomings in equipment. Be ready for moments of drama or beauty. Also try to snoop around and be a little daring.
A couple of my favorite F1 photographers are James Moy and Darren Heath.