Although advice is easy to come by on this site, you definitely want to do some outside research for this sort of purchase. dpreview.com is a great site with reviews on most DSLRs as well as many lenses. And if you do purchase a new DSLR, I highly recommend David Busch's Digital Photography Guide (for your particular camera). I had a Nikon D100 for about 8 years, and just stepped it up to a 7D, and his guide was great. Since I'm now Canon guy, I'm going to go ahead and tell you to look into the t2i or the t1i. At least between Canon and Nikon, I think they are the only options out there with 1080 under $1000. I believe
they have similar video capabilities, but this is something you should check out.
With all DSLRs, they are offered either body only or in a kit with a lens. For the cameras I mentioned before, you can get either an 18-55mm f/3.5-5.6 or 18-135mm f/3.5-5.6. The problem with either of these is they aren't very good at all for filming in low light, however they do give you a decent range if you're only filming during the daytime. Even though they are ok lenses to start out with, you'll probably end up wanting to replace them down the road. In the end, it can be better to get a cheaper body and more expensive glass, learn that camera, and then upgrade the camera itself later on but still have your lenses.
Another option is to get something like the 50mm f/1.8. This lens is only $100, gives you great depth of field, and will perform well in low light situations. The drawback here is obviously that you have no zoom at all, so you really have to move yourself around to get the exact shot you want. Despite its limits, you won't ever grow out of this type of lens. There will always be a time when it could come in handy, so it might make a good starter lens for you. There are other prime lenses out there (fixed focal length) that are a little more expensive, but will still be a cheaper alternative to zoom lenses (If you're filming skiing, 50mm will probably be too long for follow-cam stuff)
Hope this was semi-coherent and helps a bit. Good luck, and let me know if you have further questions.