Depends on what you want to shoot
At close range, or if the possibility of missing and hitting an unintended target downrange is a consideration (ok, it always is, but you know what I mean), a shotgun is the way to go. A .410 will handle small predators, a 20g or 12g is a better choice if you have larger predators, vicious feral dog packs, or the like. If there's a problem at the barn at night, grab the 12. I have had .22 rimfire go through targets at close range, with minimal bullet expansion and hence less immediate results ... a consideration when heading out to confront the unknown at night ...
If you need to reduce some problem animals at range, a .22-250 or .223 are good choices. Both are highly accurate, flat shooting, widely available, and pack enough punch for anything up to coyote. The .223 is primarily a military round, but you can find both sport rifles with bolt action, or get a Ruger mini-14 or AR15 in this cartridge. I'm not a big fan of the .17 being touted here, at least o=for your purposes; true, it is accurate and flat shooting, but so are the .22-250 and the .223, and they pack more punch with no real downside trade off vs. a .17 short of the .17 being quieter. Either the .22-250 or the .223 are far better for coyote and such. There are plenty of other options with similar characteristics, but unless you're making a hobby of your firearms, there's no need to pay more for more esoteric ammunition.
If you have very large predators, the old standby .30-06 is hard to beat. Widely available, and if you're going to own only a single larger-game rifle, this is the choice, though there are plenty of other options of course.
As small handguns go, Ruger's .22 is inexpensive and well made. Personally, I like my Hammerli SP20 and it's 1/2" groups at 75 yards, but unless you're match shooting, there's no need to spend that sort of cash.
... the older High Standard .22 are also nice, and can be found at a good price for the accuracy if you want to go retro ...
If you need a larger handgun for larger, and posibbly 2-legged threats, go with a .357 magnum or a .45. The .357 magnum has the advantage of downsizing to cheaper .38 special ammo for cheaper practice and more versatility, and a .357 w/ 6" barrel is accurate out to 100+ yards if you need something for open terrain. I have an older Colt Trooper MKIII 6" and it is great as long as you don't mind stapping that weight on.
Kimber makes excellent .45 autos, and they have a "small hands package" that might be a great fit for the OP, since she said she has small hands. However, any of the "1911" style frames can be outfitted with what's called a "short trigger" inexpensively.
This certainly seems to be a popular thread; I suppose raising chickens and keeping firearms around both hit that same self-reliant streak.
