If recoil is really an issue to you, Remington makes a Low Recoil line of ammunition for small or novice shooters. Ballistics are still comparable to the .30-30, IIRC, which is plenty for most North American game at ranges they are typically encountered.
I don't think the .30-06 should be a problem with proper care taken to good fundamentals. I started shooting the .30-06 back when I was 11 or 12 years old. It took me by surprise at first, but I quickly adapted and it never hurt me. That was an old 1970s M700 with absolutely no sort of butt-pad at all. Modern recoil pads help soak a lot of the hurt. Make sure you hold the rifle tight and have enough eye relief on your optics, and take it slow. Don't try shoot 30 or 40 rounds on your first range outing. The minute you start to feel sore or catch yourself anticipating recoil, put it away and go back to the .22. Over time, your body will learn to deal with it and slowly, your mind will comprehend that even if the sensation isn't pleasant, it isn't exactly painful either.
I would recommend you invest in a .308. It should be plenty for deer, elk, moose, and black bear if you keep ranges sane. You could actually reach out and touch a deer or black bear from a ways but I would recommend you not until you know you have the patience and skill level to accomplish it. And though some may disagree, I wouldn't be shooting at elk or moose with that cartridge much past 250 yards.
And I'll let you in on a little secret...
Cartridge selection doesn't matter nearly as much as some think it does. There are a whole host of cartridges that will work, and game animals won't tell the difference. The cartridge you choose isn't nearly as important as the bullet you choose or acknowledging your own limitations. If you just don't like recoil, or you have a shoulder or back injury, or other handicap, a .243 or 6.5x55 will do you far better than a .308, or even a .338. I know someone who killed a lot of elk with a .243 and 100 gr Corelokts. I know, it's not what I'd call ideal either. Apparently, he didn't get the memo, and neither did the elk. He was a very accomplished rifleman with the patience to wait for the right shot and the discipline to pass the shots he couldn't make. And he never went hungry for it. The Europeans have used the 6.5x55 on moose for decades and it reportedly does just fine. I am a big fan of using enough gun, but for me, this means using the most gun that you can shoot comfortably. Shot placement and bullet selection are far more important.