RE: .308 or 30-06
30/06 - Win. 1895, pretty nice but real pricey, and BLRs.I don't know of any others.
308 - Savage 99's and BLRs.
I think you can find used 99s, but they can cost a bit. Some of the lesser quality ones may be cheaper, and they all seem to shoot well.
If you like levers, you should also think about Marlins and Winchesters in various other calibers. I'd buy the first goodMarlin I found in something bigger than 30/30. 35 Rem., 356 Win., 444 Marlin, 375 Win. are all good as far as I need to shoot.