Well, like everybody else and for the same good reasons, my first recommendation is to get a 22LR. Any 22, really, as long as it's reasonably accurate! Still, if you're dead-set against a 22, consider my formal recommendation lodged and let's move on.
What other types of rifle you pursue depends on what you consider fun and how far away you want to shoot. Since you're just going to be shooting for fun and not hunting, that puts us into two different possible territories. First and most relevant since you already have some handguns, would be a pistol-caliber carbine like a Beretta CX4 Storm, a Kel-Tec Sub2000, or a Hi-Point Carbine. They're accurate, more powerful than a 22LR, and make good self-defense weapons to boot. They're safe to use on pistol-rated metal targets, and satisfying as all get-out to shoot. You can scope them or red-dot them if you like, or just roll with the iron sights. Either way, they're a barrel of fun and cheap to shoot.
The other way to go would be a bolt-action like you hinted at when you mentioned a Remington 700, which is sort of the generic American modern hunting rifle. With a 700, you're almost required to purchase a scope and rapid-fire is definitely out, but that type of gun leaves your caliber choices wide-open. If I were going to be shooting a bolt-gun for fun and hunting (particularly deer, because of caliber limitations) wasn't an issue, I would get any quality .223 Remington and have a field day. Low recoil and flat shooting make it easy to use, and cheaper ammo than other options increase the number of rounds you get to shoot before pain sets in. If you want about the same amount of power with a larger bore and slightly cheaper ammo, consider getting a CZ 527 in 7.62x39. It has the correct size bore for the inexpensive ammo, and if you decide to handload or purchase premium ammo from manufacturers like S&B, you can get some serious accuracy out of it. The CZ 527 comes with decent iron sights and provisions for scope mounting so, you have the best of both worlds. For what it's worth, the 527 also comes in 223.