First... realize that sports photography (fast far away objects in challenging light) is difficult and equipment to support it can be extremely expensive. This is why professional sports photographers are well paid. WIth that said, it is very possible to obtain good results with in a budget... You just have to realize and set expectations. If your typical outdoor lighting is good (bright), then your options are wider.
You will most likely need to use a monopod (or tripod) to help steady yourself. You should work that into your budget as well.
In that budget, I'd first start looking at third party lenses and decide on the camera afterwards. For example.... Check out either... (I'm sure others here can suggest others)
1) Sigma 170-500mm 170-500mm f/5-6.3 APO DG about $770 USD
2) Sigma 135-400mm f/4.5-5.6 APO DG $590 USD
3) Tamron SP AF 200-500mm f/5-6.3 Di LD IF about $900 USD
4) Tokina Telephoto AF-D 80-400mm f/4.5-5.6 ATX 840 $650 USD
do your research .. ask around here. Once you have made your choice, then go to a camera shop and hold camera bodies for which the lens manufacturer can accomodate. Your choice of body should be based on what is left in your budget as well as what feels comfortable... Then order the lens that fits that camera and the camera itself.
Don't forget memory cards too.
Last but not least... try to squeeze in a wide to medium zoom for other shots. Don't forget to investigate used to squeeze more out of your budget.
Btw.. you never said if you were interested in digital or film.. I'm assuming digital.
