One of the best options for a hunting rifle imo is the Savage package deal they sell at Walmart. You can get a long or short action. The short actions are Model 10's and the long actions 110's. They feature the famous accu trigger which is probably my favorite factory trigger I've ever used. It's a great feature to have. They run right under $400 and I just don't think you can beat it for that price. Some of them come with Simmons scopes that are decent and should get you shooting, but something you will probably want to eventually change. The one that our store got in recently has a Bushnell scope on it. I've used the same Bushnell scope on another gun and it's my least favorite scope I've ever used, so I was sad to see them change them. However, it should still get you shooting right out of the box.
Then I'd buy a quality set of steel bases from any of the major problems. For rings I'd go with the Burris Signature Zee rings. They are my favorite I've used for under $100.
Then when it comes to scopes there are lots of great options out there. Honestly the Zeiss Conquest for $400 in 3-9x40 is very very hard to beat. You could buy this and the rifle, mounts, etc and still have $100 or so left over for ammo. This is the way to go imo.
If you don't want to spend that much on a scope, I'd pick up one of the Vortex Vipers you can still find on clearance. I really like the 3.5-10x50 that Cameralandny.com still has on clearance. Another good option is the Burris Fullfield II scopes. They are great scopes and I love mine, but the Vortex is a better deal right now as you get even better glass and an even better customer service department for the same amount of money. When the Vipers aren't on clearance I think the Burris Scopes are one of the best choices for the price.
SWFA had some of the Burris scopes on clearance also. I'm not sure what they still have left. They did have a 3-9x40 for $120 and a 4-14x42 for $200. These would also be great choices.
That would give you a great hunting rifle with a great optics and still give you money left over to buy some ammo.
Another entry level rifle that's been getting good reviews is the Marlin XL-7/XS-7. I've checked them out and like them, but I prefer the Savage to the Marlin. I don't think you could really go wrong with either.
One other rifle I really like is the Browning X Bolt Medallion. I'd like to pick one of these up eventually, but who knows if it will ever happen. The problem with this is they run about $850. So by the time you buy bases and rings, they doesn't leave you hardly any money for optics. You could go with one of these and one of the Nikon Prostaff's on sale right now and the bases and rings mentioned about and be right at your $1000 budget, but once you added tax in you would be over $1000. This would be a nice setup and decent optics, but I'd honestly rather have a Savage with great optics if it was going to be a hunting rifle. The Savage will be just as accurate if not more accurate.