I'm not a hunter, so I won't give you any advice on that front. But I do own a 300 win mag rifle, so I can give you some general advice.
First, this ammo is expensive. If you've been shooting nothing but .22 you're in for some serious sticker shock. So this is not the best ammo for plinking. Honestly, I think you'd be better served with .308 of 30-06, both of which will easily get you out to 400 yards and can be had more cheaply.
Second, if you've shot nothing but .22 it's going to take some time for you to adjust to the recoil of 300 win mag. I shoot this ammo out of an AR30 with a muzzle brake, so recoil isn't too bad, but this is a very heavy gun. It's not something you could use for hunting, or really even shoot offhand. I have to believe shooting this ammo out of a relatively light rifle without a muzzle brake is going to have some serious recoil. Yes, you can get used to anything - but it's not a great choice for someone used to a .22
Third, 300 win mag is a barrel burner. Are you ready to replace the barrel after 1500 rounds? Again, this is not a big issue for me because I own a bunch of guns, so the AR30 doesn't get a lot of use. But this is not a great second gun.
Honestly, I think you'd be a lot happier with something in .308, even if it doesn't have the wow factor of 300 win mag. In fact, I wouldn't even dismiss something in .223. Ammo is cheaper. Recoil is pretty light. A 68 grain or larger bullet will get you out to 600 yards (at least if you're shooting paper). Although it may not be the best hunting round, at least for large game.
Finally, if you really want to go big bore there are a bunch of other options which I think might be more fun. If you want something in an AR platform there's .458 Socom or 50 Beowolf. Big kick. A lot of foot pounds down range. Certainly good for hunting if you don't need to go more than 150 out.
Or you can go whole hog and just get a .50 bmg. Of course, if you go this route, everything is expensive. The guns. The ammo. The scopes. The rings and mounts. But shooting one will definitely put a smile on yourself. And it will take out a mastadon at 500 yards.