Ditto 30.06 Spr. Very good all around 30 cal. cartridge for hunting. Every maker of center fire hunting ammo that I know of offers several options for the 30.06 Spr.
Take a look at the Ruger American rifle. I have seen these selling in the $300 range. But do not leave out the "good" used rifle option. You can often find a good buy on used gear. If you are new to buying center fire rifles, take a friend that has experience ... especially if buying used.
If you need (or just want) a scope, with a $600 or so budget, you will not have much left for glass after buying the rifle .... but in the $250-$300 range you can definitley find a decent scope. Look at Vortex, Leupold and Bushnell scopes in this price range. Some will disagree with me on this ... but stay away from Simmons, Tasco, and BSA. I and several of my buds have had poor experiences with scopes carrying these brand names. If you can swing it, the Ziees Conquest is a very good value, but it is going to push $450-$500. A 3-9x40mm should suffice.
Once you get the rifle-scope combo, you'll need to head to a range with a few different factory loads and determine which groups the best. LOts ofinfo on the web of how to sight in a rifle. I suggest trying out ammo with a bullet weight of from 150 gr. to 180 gr.
Good luck and welcome to the madness that is deer hunting !