G29, you will undoubtedly get many recommendations for either a Savage or a Remington; and they wouldn't be bad recommendations. I'm one of the guys from the Remington camp. I've shot quite a few Savages and they shoot nicely. I just prefer the Remington action.
When you say that you want to shoot beyond 500 yards, what exactly do you mean by that? Are you just planning on banging steel? Precision shooting? Want something that has a "tactical" edge to it?
Now, are you shopping brands or brands and caliber? I shoot a .308, but I do so because I built my rifle to shoot F Class T/R, so I had to go with .308. Well, I could have gone .223, but prefer the .308.
Another thing that you have to consider is the scope. If you want to keep the whole system to under $1000, you're going to have a bit of a challenge.
One option that pops into my head is a Remington SPS Tactical. I've seen them for around $550, if you shop around. For glass, you can get a plain jane Weaver base for around $20, a decent set of rings for around $50 and either a Bushnell fixed 10X ($200) or a Super Sniper fixed 10X ($300). It's a bit over your budget, but I think that anything less will leave you wanting.
The best option to help you stay on budget would be to scour the for sale forums and look for something used. I've sold a few rifles here and at Snipershide.com
As far as the caliber goes, the .308 is popular, but there are many calibers that are superior to it for distance shooting. If you aren't committed to any particular caliber, that may be what you should decide on first.