Winchester used to make a .30-06 M70 target rifle, with a heavy barrel, but the .30-06 has kind of fallen by the wayside with high power competitions using the .308 (and others.) The .308 has similar performance to the .30-06.
You can get something like a Remington 700 SPS Varmint in .308, and it will weigh something like 10 lbs with a scope. This will help minimize perceived recoil, compared to a lighter rifle in the same caliber.
You could also get something like the Savage FT/R (also .308), which weighs 12.5 lbs before you put a scope on it. With a scope it would probably weigh 14 lbs, which will definitely make recoil more bearable.
If you don't like the factory recoil pads, change them for a "high end" one like a Limbsaver or Decelerator.
Just my opinion, but if you're doing target work and not hunting, you should look for a heavy barrel model such as the two mentioned above.