Here in OK where I live, wild dogs and boars are the greatest risk. Further South, close to the border, two-legged varmints are an issue. In CO, where we ride in the summer, bears and mountain lions are a concern. I generally carry a single action Ruger New Model Blackhawk in 45 Colt strapped to my hip using a Triple-K Wyoming Western Drop-Loop Cartridge Belt Holster; I also have a Western Express leather holster tied to my offside rear billet: If I plan to do a lot of loping, it’s hard to get a belt holster to not bang into the saddle; also, in rough country, left-side right leg forward rapid dismounts or reins can tangle with the hammer with a hip holster. I don’t recommend even double action revolvers or heaven forbid, a semi-automatic with a light trigger pull; it’s just too easy to fire inadvertently. Also, be aware that with a revolver, hot gas and unburnt powder fragments fly out at right angles to the revolver’s forcing cone, just in front of the cylinder, and that can injure a horses eye. So you should be shooting to the side, not forward. Jerry Miculek has a YouTube video showing what happens to a hot dog or finger laying in that area. I use hardball 315 grain lead bullets and hot loads in CO for better penetration, and less punishing 250 grain copper plated loads for dogs and boars.
On occasion, I also carry a 45 Colt lever action carbine in a scabbard, but I never fire off the horse with that weapon. Regarding training; it took quite a while, maybe a year to graduate from hand claps, cap and starter pistol, primer only then gradually increasing powder loads, first on the ground and then in the saddle, initially at the whoa, then at a walk, then at a trot, and then shooting to the left (I’m right-handed). Oddly enough, it’s a tie between my spookiest (named Spook for some reason) vs. my most laid-back horse of the three I ride, who became the best at it. The training crosses over to general desensitizaion, as well. Also, firearm training is a perishable skill for horses as well as people; it needs to be revisited at least every month, IMO. I have a few cones set up with balloons we zoom by and kill at the beginning of the ride, when the horses are still fresh, which the horses seem to enjoy.