Remington 700, Winchester 70, Ruger 77, Savage 110 to start with. There are numerous variations of these as the makers catalogs will show you. Any are capable of 1 minute of angle accuracy. Some will do better.
If econony is paramount, some of the military rifles on the market will show acceptable hunting accuracy, if the barrel is in good condition. Some of the Swedish Mausers in 6.5 x 55 are superbly accurate. All will need a lot of work to make them the equivalent of a modern sporting rifle.
If you want a sporter rifle, one which is light, handy and which will take a scope, go with one of the U.S. made rifles above. I would suggest going to area gun stores and picking up some flyers. Talk to some dealers and see what is available and handle some rifles.
Then, when you are ready to buy, remember the dealer(s) who treated you right. You may save a few bucks buying from a discount store, but when you go back for accessories or service you will find that the clerk doesn't know beans about guns. Buying "on the net" is even more a crapshoot and you will have to take delivery through a local dealer anyway, and pay a service charge.
Jim
Any suggestions on decent rifles that are in the 500-600 range or cheaper would be appreciated too.