I have a Finnish Mosin, and they are usually the most accurate of the Mosins. If you replace the sights with Mojo sights for around $60, most of the time you can do 2" groups at 100 yards with them. Russian Mosins tend more toward 3-4", with some happy exceptions. Russian Mosins can be had in my area for about $130. That's hard to beat for a rifle in the 308 to 30-06 power neighborhood. The problems are that they are heavy and that mounting a scope is not very easy.
I would tend more toward picking up an inexpensive modern bolt action rifle, either new or used. With some looking, you can get something with a scope for around $350 or so. For just exploding pop cans, breaking rocks, and target shooting the 223 is very hard to beat. If you're going to hunt deer, then pick from 243, 260, 7mm-08, or 308. If you get a 30-06, you probably won't enjoy doing a lot of shooting at the range, but you'll be able to hunt anything in North America.
For basic fun, you might look around for one of the old Savage or Stevens bolt action 30-30s. Those can be had for $200 or so. They are mild, fun to shoot, and potent enough for deer out to maybe 125-150 yards. The other really fun alternative is the Marlin lever action in 357 Magnum, the handgun round. Those are extremely fun to shoot, very mild, and still enough for deer out to 100-125 yards. If you reload, they are very cheap to shoot. But those will run about $400. In 22LR, take a serious look at the Henry lever action rifles. They run about $260 new and are a lot of fun. They are big enough for rabbits, skunks, and raccoons, very mild, and a nice firearm.