A good .22RF and a good .308 is an excellent basic battery. If you want a rifle for both target and big game you can't go too much lighter than .308. Yes. 6.5 X 55, 260, 7-08 and .270 will all do but, other than the .270, the ammo is not as easy to find as .308.
Regarding cost, you won't be shooting 100 rnds at a time like you probably do with your .22. What I suggest is that you start a shooting session with .22 for a warm up then switch to your centerfire rifle and fire 20 well aimed shots. That won't break the bank. Save your brass for when you start to reload!
Given your stated purposes, your desire for moderate recoil and ammo availablity/cost, I think .308 is your best choice and .270 would be your next best. In my opinion .308 is one of the best all around calibers, I think I have about 6 at the moment.
A Savage is probably your most available left hand bolt gun.