My top choice would be the Tikka or CZ heavy-barrel .308's, with Savage as a second choice (and top choice among US manufacturers). If you shop around, the first-named rifles should set you back $600-$750, and the Savage $500-$600. You'd then have to use the remainder of your budget for a scope, bases and rings, a bipod or other mount, etc. For varmint shooting, a higher-powered scope will be necessary, and buying something good in the 20x range might cost you more than you have left...
However, the .308 isn't renowned as a good prairie dog cartridge. It'll do the job, no question, but a higher-velocity, flatter-shooting round like the .22-250 might be a better choice.