30.06 has more choices in loads than practically any other rifle caliber.

7mm has more power and range.

For example, years ago I was using, as a backstop, an old peppermint still boiler (abt 6' diameter x 10 ' high, lying on its side) with 1/2" steel sides full of galvanized 2" pipes. The 30.06 rounds were deflected directly into the ground. The 7mm rounds went through the front side and all the galvanized pipes before being stopped by the far 1/2" steel wall. Had to quit shooting at it with the 7mm because there was a good chance of one going all the way through (yeah I sometimes put more than one round through the same hole).

If you reload, the cost is similar. If you just shoot the cheapest store-bought ammo, then the 30.06 will be cheaper.

If you are only going to be target shooting, you might consider a smaller/cheaper caliber, such as an AR-15 .223 cal with a drop-in conversion kit to allow you to shoot .22 cal. Despite the frequent shortages, it is still far cheaper than practically anything else.

As for the .308, with the longer barrel it does make for better accuracy at shorter ranges (for the infrequent hobbyist, at least). At longer ranges, especially with windage, the 30.06 or 7mm have it beat. Many also consider the 30.06 a better choice (accuracy-wise) if shooting anything over 200 gr. Sure, people may point out that the military uses it, but the 30.06 was also a highly respected military weapon for several decades. Some think that it is easier to obtain match-grade ammo for the .308, but serious shooters don't consider any store bought stuff to be "match-grade" anyway.

Now if you want to have some serious fun, check out the Kel-Tec sub-2000 .40 cal or 9mm with a glock 22 or similar magazine. Folds down to a 17" length so it is easy for travel, too. Of course, it helps if you save the brass and reload.