Lots of good advice. One thing to keep in mind is not only the focal length (ie 300mm), but the f-stop range. The more "powerful" lenses cut down on light, and the lower fstops will allow more light, and faster shutter speeds. (to stop action) You will quickly notice you can get many high-power zoom lenses relatively inexpensive, but try finding the lower fstops, and the price skyrockets. There's a good reason. Quality glass (or most any glass) in the bigger diameter get expensive quick. You can get a very good Nikon 70-300 zoom for around 500, at f4.5. Want a f2.8 and you're gonna pay around 5000 for a 300mm!
The lower fstops are often referred to "fast glass" because with their superior light-gathering, they allow for much faster shutter speeds. The "long-fast glass" like you see on Nat Geo shoots often run as much as a decked-out Gator XUV.
One BIG, no HUGE advantage of D-SLR cameras over the point-n-shoots are the shutter lag times. I love my Canon shur-shot, and it takes fantastic pictures....of anything that's not moving! You push the shutter, then wait up to a couple of seconds for the camera to expose. By then fido is out of sight and you have a picture of his footprints. The DSLRs will reconstitute much faster also, so you don't have to wait for the second shot.
All said and done, you should be able to get a good kit camera from Canon or Nikon with a decent zoom lens in the "prosumer" range for around 1000. Used would probably put you in the 800 range. Canon vs Nikon is like Ford versus GMC. (with Nikon being comparable to the vastly superior GMC!)
www. nikonian.org has more info and helpful amateurs and pros than you can imagine. And they don't have a POTUS room to suck you into!
Good luck, but be careful, photography can become an all-consuming passion, just like dogs,
Dave