That budget should be able to get an entry level D-SLR and a good lens.
For the D-SLR, entry level will get you continuous burst at 3 fps. The mid range D-SLR like Canon 30D and Nikon D200 can let you shoot at 5 fps.
For the lens, you need to decide what focal length you need and what lighting condition you shoot at. If you always shoot outdoor in good day light, you just pick the focal length you need and don't need to worry about the aperture as much. If you sometime shoot indoor and in low light, you need a fast lens with aperture f/2.8 or wider (the smaller the f/number the wider the aperture, eg. f/1.4 is wider than f/2.8).
Canon Rebel XTi (3fps) or Canon 30D (5fps)
In good day light at close range: 17-85 mm IS or 28-135 mm IS
In good day light at longer range: 70-200mm f/4L(IS) or 70-300 mm IS
Close up at lower light: 17-55 mm f/2.8 IS, 50 mm f/1.4, 85 mm f/1.8
or Tamron 17-50 mm f/2.8
Longer range at lower light: 100mm f/2, 135mm f/2L, 70-200mm f/2.8L,
or Sigma 70-200 mm f/2.8
(The 70-200 mm f/2.8L IS is good for low light action but is slightly out of your budget).
For long distance: 100-400mm f/4.5-5.6L IS or get a 1.4x or 2x lens extender. The size, weight and price are insane for lenses with focal length longer than 400 mm ($5000 plus). The only lens with 500mm within your budget is Sigma 170-500mm (but don't expect too much at the 500mm end, and only use this lens in good day light outdoor with a tripod).
Outdoor and don't want to change lens: Tamron 18-200mm or 18-250 mm
For Nikons and other brands, I let others to give you more info.
You can check prices at www.amazon.com or www.bhphotovideo.com
I am looking for a good camera that has a very low shutter lag (preferably none!), good image quality, and the ability to take good pictures in both light & dark settings. The camera would be used for action shots and would need to cost less than $2000. I currently own the Lumix DMC-FZ7 and it just does not seem to be fast enough for the type of shots I'm looking for.
If anyone has any suggestions, I would greatly appreciate it! Thanks!