Fun? Target? Hunt?
Yeah, that's kind of like asking what next auto....
But, for a 'new guy' to firearms, assuming you want to play at the range with it, .223 and 7.62 x 39 are good choices. Ammo for both is cheap and available everywhere. You can get a Savage bolt action for the .223 without breaking the bank; you can get an SKS or any of its variants for even less.
You can hone your skills as a shooter with the .223 without developing flinch; you can play at the range with an SKS having fun with paper targets. The .223 would give you an excellent beginner's varmint gun, while the SKS would give you an excellent truck/trunk weapon.
If you really don't have much experience with high powered rifles, I'd stay away from the 308 just now - and certainly anything bigger - because now's the time to build your basic skills. Nothing wrong with the big bores, mind you, but it's a big leap from the .22, and I think your first high powered rifle should be more of a 'fun gun' that you can shoot a lot, to get very familiar with what they can do.
That's just my OPINION. Your results may vary. Have fun.