Take your pick, nearly every manufacturer has a rifle for under $600. I'd say remington because I like them and the SPS is accurate out of the box, sub .5 MOA if you change the stock. Someone that likes savage will say savage because they're accurate, someone that likes CZ will say CZ because they're accurate, someone that likes winchester will say winchester because they're accurate.
Caliber-wise, I'd say 308, but someone that likes 30-06 will say 30-06, as will someone that likes 243 or 25-06 or 223 or what have you. 308 is a good choice as it is widely available, moderate recoil and good ballistics for the ranges you will most likely be shooting at. It's an excellent do all cartridge. Plan on getting a .223 in the near future as well.
Scope: if you're planning on getting one, you should really try and get something like a Leupold vari-x III (can be had for $600). The glass and the warranty are superb. You will be able to find a scope for much less, even around $40; but you get what you pay for and $40-$100 scopes are cheap for a reason. Buy something you won't have/need/want to replace.
Edit: Do not get the 770 or any rifle with a scope package. It may look like a good deal but it's not for the reasons I stated above. Don't go el-cheapo unless you plan on shooting this rifle 15-20 times a year. If you plan on getting into shooting, get a bolt gun. I can't really tell you about any other rifles in your price range other than the remington 700 SPS which I have read several reviews on.