I'd consider shooting a cougar to be a last resort; scare it off if you can. However, if you need to shoot one, and since that's what you actually asked about:
A shotgun will work fine at close range, and would take less practice to be able to use effectively. There's no reason why an adult of any size could not handle a 12 gauge; I had one when I was 14 years old. If you do find a 12ga too large for your liking, a 20ga will work fine for cougar as well. Get a short-barrel open choke self-defence weapon, load two rounds of buckshot and then alternate slugs and shot. Another advantage to a shotgun is that the rounds will not travel so far downrange, which can be a consideration. Shot will travel a shorter distance than slugs, but either will travel less distance than a rifle bullet. If you're smaller, you can get a self-defense model with an adjustable stock, or get a gunsmith to cut the stock down for you if you are not comfortable with that sort of a task.
A shotgun's fine if there's a cougar at the barn door and you need to kill it. A shotgun's not a good choice of you have a cougar that's been hanging around and you determine that you need to hunt it down and kill it. You're not likely to get close enough to use it without using dogs for assistance.
If you opt for a rifle, anything .270 or larger is fine. As others suggested, you can get used .30-30 or .30-06 rifles inexpensively, as they are rather common. Again, as with the shotgun, there's really no reason why an adult would not be able to handle a .30-06. Handling recoil is largely mental; it does not require size or strength.
A rifle's the choice if the cougar must be hunted down.
A handgun has the advantage of being easier to carry about with you, if that is legal where you are. On the other hand, using a handgun effectively at anything other than extremely close range requires more proficiency than using a shotgun or rifle, and a handgun will have somewhat less stopping power. I would recommend a .357 (has the advantage of using cheaper .38 special ammo for practice) or a .45. Revolvers are somewhat easier for most novices to learn to use, in my opinion. Again, handling recoil is a matter of training and practice, not physical size.
The general rule is that the best gun is the one you can hit your target with. One could go on all day arguing the relative merits of one cartridge over another, but the difference between practicing or not far outweighs any difference between a .270 and a .30-30. Get something that fits you well and practice with it.
If you are new to firearms, get proper instruction. Your state likely offers a hunter's safety course, and you can check with your local firearms dealer or sportsman's club for local resources. You can also check for NRA safety courses.
