Any of the '06 based cartridges will fill the need.
25-06,270,280,30-06,,with the 30-06 being the most commonly avail ammo in any store ranging from a mom and pop country store to the largest sporting goods place.
Other good options would be the .257 Roberts,308,7MM Mauser,6.5 Swede all based on an intermediate action,a little wt savings if that matters.
For deer and Elk,hard to beat the 180GR '06.
We have most of the calibers i mentioned above,they will all do the job quite nicely,as long as you do yours.
If you're not into hand loading...Get there! Remember, the lighter the bullet, the lighter the recoil. And if you really want a "Trainer" Check with some of the older guys for a "Marble's Sub-caliber Adapter". You clipped a .32 Auto cartridge onto the front of it and chambered it into your 30-06. No recoil and it was great for killing tin cans.