I don't own a bolt gun, but shoot lots of them. Guys out at my range are really generous and I've been looking for a bolt gun myself. IIRC, the difference between the .308 and 30-06 is about 200 fps. Not that big a difference. In like rifles, there isn't that big a difference in recoil either. Remember, I don't own a bolt gun, but have shot a bunch. What this means is that all of my shooting with them has been done in comparison and I haven't gotten used to anything.
I chose the .308 due to a few considerations. First, how often is the rifle going to be shot? If you're like my uncle and rifles only get shot when there's a deer in the scope, it may not see more than 500 rounds in it's lifetime. If this is the case and you want to maximize your range, go with the 30-06. Good, accurate factory ammo is readily available. Again, in this case, buy the rifle and not the caliber (if maximizing distance isn't a concern). Either caliber will do, so just find a rifle that you like.
If you're more like me and will shoot the rifle a LOT, like at least 500 rounds a month, there are other considerations. I'm a reloader, so for me, the considerations are availability of brass and cost to reload. .308 is going to be cheaper because brass is plentiful. If you're not a reloader, but will still be shooting a lot, .308 is going to be easier on the barrel than the 06 will be. If you are a reloader, you can load the 06 to do what ever you want, but there's still the cost to reload issue, dependent upon how much you are going to shoot.
Just think it out and fall on what fits you the best. When comparing .308 to 30-06, I doubt that either one could ever be the wrong choice, although one may be the better choice for you.
I went with .308 and am now shopping for that deal that I just can't pass up, because it's going to be an easy transition. Cheap to load and finding a load that will work for what I want will be easy.