Scopes are for pussies.
Get a surplus or immediate post-war WWII bolt action. I recommend the Mauser Gewehr 98 family. Reliable, simple to use, excellent saftey cam, and zero risk of slam fires. German ones will usually be in 7.92 x 57 JS, sometimes called Mauser 8mm, but you can find post-war cousins chambered in dirt-cheap and easy to find 7.62 x 51 NATO. However you lose stopping power with the lighter NATO round, so it really depends on if you want a piece of history that can shoot through the better half of a tree, or a somewhat less powerful rifle that doesn't cost 25 cents a pop to shoot. I'd recommend picking up a bastardized, I mean "sporterized", original Mauser or one that's otherwise better suited in a collector's hands. Keep in mind that shooting gators will likely be done at shorter ranges, so I don't see a scope being too important, but you need to be able to comfortably fire the weapon standing or from a tree. However you probably can go too big shooting gators if you're shooting at closer ranges, so I must advise againt the bigger calibers. You'll be overpowered for rabbits (unless you're okay with wasting a carcass), coyotes will be easy game as far out as you can shoot them, and turkeys should also be managed.
However I'm a target shooter, not a game hunter. Your best course of advice is to go to say a deer butcher, game taxidermist or other similar hunting-related business and ask the most knowledgable-looking guy what he'd recommend. I wouldn't ask a gun store clerk, although the patrons would be willing to give you an honest opinion. Ask your older male family members. But above all, scopes are for pussies, guns are serious business, and a surplus service rifle is a cheap way to get a shooting iron you can't break without the application of outright malicious stupidity.