Seriously if bigger game could be on the menu I would look at the
.270 Win, .30-06 or .308 Win. You can learn to tolerate the recoil, if I were you I'd start with a PAST recoil pad on the shoulder. I started out this way, and now I only use it on cartridges more powerful than .30-06. Practice shooting the .30-06 first and then go to the .22 for the rest of the day, don't beat youself up trying to shoot a lot of .30-06 ammunition.
When I go to the range to practice I usually don't shoot more than 20 rounds out of my .30-06 and spend the rest of the time plinking with lighter cartridges. I only shoot that many rounds if I'm doing load development, other than that it only to confirm zero one or two three shot groups and I'm done. The mechanics are the same shooting the .22 as the .30-06 so you will get a ton of good practice. Learn to handload as mentioned to add versatilty to your rifle. You can go from mild plinkers to heavy shoulder bruisers.
However, reading most of you post you should get a .243 and .30-06 then you can hunt anything you wnat in the lower 48 easily.