When you said you have a .22 did you mean a .22 rimfire? or .22 centerfire?
That sub 500 dollar remington comes in the .260 rem and the 7- 08 rem. Both have very low recoil and the capacity to kill elk if you ever go that high. Both have scads of bullets available, and can use almost every rifle powder made in one way or another. ALL remington triggers are adjustable, and the way to do it is easily found on the internet. I have several of its predecessors , the 700 adl synthetic, and love them, all shoot just fine as well.
One thing about long action to short action is the cost of powder. A long action using an 06 case, (25-06, 270, 280, 30-06, 35 whelen) is going to use about 54 to 56 grains of powder per shot. That equals about 130 loads. A short action generally uses about 44 grains, which equals about 160 rounds...so you get 30 loads per pound of powder for 'free'.
WHile an 06 is a wonderful cartridge, and it does have lots of versatility, its downsides are recoil, more expensive ammo, a longer rifle, and the need for a longer barrel most of the time. Shooting a .260 would save you on powder, weight, recoil, economy, and I think its more fun to shoot when you do not get so beat up.