One of the things I learned way after the tutorial was the fact that, when galloping on your horse, timing your button press with your horse's gallop won't drain your horse's stamina nearly as much, if at all. Get into a good rhythm, and you can absolutely cruise along without
much of a care in the world. This is especially useful when chasing somebody on horseback since those can end up taking an extended bit of time. The one thing to keep in mind about this is that it's pretty easy to get numb to the speed if you're at it for a while, and going through a town at breakneck pace is a pretty easy way to get on people's bad sides.