NETim, Welcome!!!!
Just a FYI, but the Ar-15 should not be your first choice for what you want to do.
What you should be looking at is a bolt rifle in 22-250 or 220 Swift. The 223 lacks the ass to be pushed out over 300 yards, and the wind will really dance the slow 223.
Your better choice would be a Rem 700 VSSF to get you started, topped off with a 6x18 power scope. Handloaded, the rifle will shoot under .5 MOA verse the 1 to 2 MOA that the ar-15 is going to shoot. That means at 300 yards, the AR will be grouping over 3" plus wind drift, which is not going to work for a prairie dogs.
Don't get me wrong, I use ar-15's for coyote hunting, but not for the first shot. I use the rig for running shots, where I can walk the rounds in as the coyote is high tailing for the next time zone. It isn't pretty this way, but does the trick.
Bottom line: REM 700 VSSF in 22-250, fitted with a 6x18 power scope. My choice would be a Leopold, but this may be out of your price range.
If your not set up to reload, then start adding this to your budget as well. My choice would be a RCBS master reloading kit, with neck sizing die kit, instead of full sizing dies.
If you play your cards right, you could have the Rem 700 rifle,scope and reloading kit for the price your going to spend on just the AR rifle. Which, would make you a serious threat to the varmint population, instead of just spraying the plains with a semi auto rifle.
Food for thought.