I have two followup questions:
- How long does it take to be at least basically proficient at shooting?
- If I were on horseback, I'd be shooting with one hand, right? Because the other hand would be holding the reins? Would I expect my accuracy to be pretty low because of that? But maybe, in the situation envisioned, it isn't really accuracy that's important, but that the object of my concern realize that I have a deadly weapon?
Sorry that second one is really more than one question LOL.
#1 I can only say, it depends. You can take a 4 hr basic hand gun safety course and at least find your way around your pistol. You'll do a little shooting after the lecture portion of the class. (Your class may be longer, in CA it's an 8 hr class and required.) A lot of ranges offer women's only classes, I really recommend that for you and your daughter. Depending on her age, she may need a youth class rather than to go in with adults.
Once you've done that, then there's Hand Gun Safety 2, some places may just do it as part of the basic safety course. You do more shooting and learn more about where to shoot, how to group your shots and do quite a bit more target practice.
Once you've taken the basic class, the sky is the limit. You can take the Concealed Carry Class (not sure what it's called or if required in TX), you can take advanced training with hand guns, guns with sights, guns with laser sights, night shooting, simulator classes that put you in 'shoot don't shoot' scenarios, rifle classes, shotgun classes, Holster classes, just about anything you can dream up, they'll teach you.
You can get involved in competition shooting, that's fun too.
So, it depends on how involved you want to be, how much time you want to devote, how often you take time to practice at a range on your own.
#2 Yes and no. Yes, probably one handed if it's an animal and you want to scare it off. No, if it's a human or an animal and you want to be somewhat accurate, especially in the beginning you need both hands. In all the training I mentioned above, you should have practiced with 2 hands, 1 hand, shooting with your off hand, and how much recoil your gun has. That one is HUGE. I love my Ruger .357 with 6 inch barrel. I mostly shoot smaller ammo in it though because it kicks like a mule. One handed, if I don't practice A LOT, I might end up shooting the end of my nose off. Both of my Glocks, the 17 & Baby Glock 26 are much smoother, but still, you need to practice one handed shooting.
If you want to live through the experience, you should practice desensitizing your horse to gun fire. The saying, "You can shoot off of any horse. ONCE." is because if you don't, you're probably going to go for the ride of your life when that gun goes off and you won't want to shoot off of him again. Most horses can be desensitized enough to be safe, if not world class mounted shooting horses.
Your last question, assumes that your target is a human and can recognize a gun. A hog, a dog, and bear or mountain lion, don't recognize such things and furthermore aren't impressed, until all the noise and stink of gun smoke starts. So, for the situations you're mostly talking about, trail riding, I'd say no.