I know it's not being pushed hard here because of your budget, but realize that when you go down the DSLR path, the better camera doesn't always take better pictures. Unless you learn how to use it, a DSLR in Auto mode is basically a big point and shoot. There are some advantages in sensor size and low light capes, but the camera alone won't give you astounding pics.
That said, I really like Canon's DSLRs. There is nothing wrong with Nikon and the others. I just prefer the interface on a Canon. You might be able to find a good deal on a used T5i, or possibly one of the older models (look for Txi, where the X is a number). The kit lens only zooms out to 55mm, so you'll want another lens. Some of the Amazon or ebay sellers sell kit packages that come with the stock 18-55mm and 55-250 lenses. They might even come in right below $1000. If you can find one with the 18-135mm and another that reaches out to 250-300mm, I think you'll be happy. WHen you look at the long lens, make sure it has IS in the lens name. That stands for Image Stabilization and you'll want it when zoomed out, especially in the lower forest light. Look for 55-250mm IS, or 75-300mm IS in the kit.
All that said, if you don't want to deal with swapping lenses or the bulk, I know people get great results with the cameras between P&S and DSLRs. Plus you'll likely get a better zoom for the buck. Don't rely on digital zoom. Pretend it's not there. That basically uses the camera to crop the pic for you and you end up with a more grainy pic. Stick with optical zoom when you're shooting.
If you want to double your budget, get a Canon T6i with the stock 18-135mm lens, the buy a Tamron 150-600mm lens for your long shots. It looks like the beast Jesse posted and weighs about 5lbs. It sure does let me get up close and personal to critters in the woods. I know this is unreasonable, though. THat's about $2k in gear, if you shop around. I like DSLRs because of the versatility I get with different lenses. That's the only reason I mentioned my hiking kit (except I own a T5i, not the 6).
Good luck and don't get so caught up in taking pictures that you forget to look around with the ol' eyeballs!