Anytime someone asks what is the "best caliber" or "best rifle", you'll always get a lot of different opinions. My suggestion is to identify the calibers sufficient for the game you want to hunt and identify what feels best to you and what you shoot well. Also, consider the availability of ammo. Unless you hand load, you may want to limit your choices to more common and popular calibers.

Some people are more sensitive to recoil than others and a magnum, like a 7MM Mag or 300 Win Mag, may not be very fun to shoot. it's subjective and only you will know what you can tolerate. I will say that people who shoot a rifle they're scared of generally don't shoot it very well.

Also, consider the range of bullet weights available for a particular caliber. A wider range of bullet weights allows you more flexibility in adjusting your ammo for the game you're hunting. A 30-06 has a fairly wide range of bullet weights. You can go lighter for deer (110 grain) and increase significantly for elk (220 grain). Not all calibers have that wide a range in weights, which is one factor to consider if you want to use the rifle for a wide variety of game.

Here are some of the most common and popular calibers suitable for deer and elk:
270 (smallest size I would use for elk)
7mm mag
308
30-06
300 win mag or wsm

You couldn't go wrong with any of those and because of their popularity ammo is generally affordable and easy to find. You can look up the ballistics for each caliber and make your own decision regarding what you think is appropriate for the animals you're hunting, your skill, and hunting style.

There are other great calibers, but they aren't as common and ammo could be harder to find and more expensive. If you hand load you may consider other options, like the 280 AI (may not be available in the rifle you want to buy).

Best of luck in your search for a new rifle and happy hunting!