My vote would actually be for three weapons, with very different uses. I know you said one, but guns are tools, and just like you wouldn't drive a nail with a pipe wrench, you really need the right gun for the job.
1) As others have said -- a shotgun. My favorite shotgun is an antique 16 gauge breech loading single shot that has been in the family for generations. It's dead simple to operate and, for a shotgun, doesn't weigh or kick much. Can't remember the manufacturer at the moment. Unfortunately, it's also my father's favorite gun, so *my* shotgun is a modern pump action mossberg 12 gauge with interchangeable barrels -- short and long, short for when I need a wide pattern and long with a choke for hitting things at a distance.
A 12 gauge with buckshot will pretty much stop anything you want it to stop short of a grizzly. My mossberg is generally loaded with one round of birdshot, then two rounds of buckshot. I'm most likely to need it for dispatching a rattlesnake or a coyote, and the birdshot will work for that, but if I need to shoot a bear or a lion, that's what the buckshot is for.
I also have slugs for the 12 gauge in case I need to put an animal down. I've been in the unfortunate situation of needing to shoot a goat that had been mauled by a dog and not having any suitable ammunition on hand. I never want to listen to an animal scream in pain like that again and not be able to end her suffering quickly. (I had just decided to slit her throat with a knife when she passed.)
We have a couple .177 caliber air rifles for varmit control. They're good for target practice (great guns to learn on), and are great on anything up to about jackrabbit size with the right kind of hunting ammo. I routinely dispatch squirrels, gophers, and the like with them. They're very accurate and very light, and the ammo is really cheap. Get one with a scope if you have vision problems like I do (severe myopia and astigmatism) -- the scope makes a huge difference.
As a side note, it IS possible, with a lucky shot, to kill a large animal with an air rifle. I've killed a large dog (GSD) that was going after my goats with a lung shot when I had no other weapon handy. I was actually aiming for his head but he was running and turned at the last second. (And then I literally bought my shotgun the very next day.) A gut shot will generally kill a large animal too, but it's a long and nasty way to die and not something I'd deliberately do. However, you're not likely to make a humane kill that way, and there's no stopping power. I just want to note this because some people think air rifles are toys and they are definitely NOT. They should be treated with the same amount of caution and respect as any other rifle.
I'd like to upgrade to a .22 air rifle, but a good one is a bit spendy. I like air rifles because they're quiet (the noise they make is about like a pneumatic nail gun), and in my area, they're legal to shoot around buildings. I can't legally shoot squirrels in my yard with my shotgun but I can with an air rifle. Even if I could shoot the tree rats with a shotgun I'd still use the pellet gun -- it's less likely to put holes in my veggies, and the noise doesn't startle the neighbors. (In Arizona, except in emergencies, "firearms" can't be fired within 1/4 mile of any sort of building or trailer, including travel trailers. A pellet gun is not considered a firearm.)
I would also suggest a sidearm of your choice. A long gun is hard to carry around all the time.
If you live in the country, there WILL be times you will want to be armed. For example, last summer, we had a (suspected) rabies outbreak here and both my father and I had our guns pretty much glued to our hips.
I carry my pistol when I'm berry picking or hiking, particularly if I have the goats or dogs with me when I'm hiking. We have bears and lions here, but I'm actually more worried about rabid critters. The human element is a distant concern and I'm not really that worried about critters coming after me (unless rabid) but I do worry about the safety of my animals. Dogs are on the menu for some lions, and everything bobcat size and up considers goats tasty.
If I could only have one weapon, it would be a sidearm of some kind because it's pretty hard to carry a shotgun on your hip, but I would REALLY miss my air rifle ...
ETA: Before buying a shotgun, be sure to try that model out. I wish I'd gotten a different model of shotgun. That 12 gauge has a heckuva kick and I don't shoot it nearly as often as I should because it makes me sore. The last time I took it out and practiced with it, I had bruises later. If I were actually hunting something that I needed a shotgun for, I'd probably borrow my father's 16 gauge. It kicks less, and it's lighter, so my arms don't get fatigued as fast. My 12 gauge is perfectly serviceable and is the "right tool" for a lot of jobs, and I just suck it up and deal with it when I shoot it (I'm a big girl, a few bruises aren't that big of a deal) but it's definitely not my favorite gun by a long shot.