Specialty Stats Help!

I need help on how to level up specialty stats. I know that training and treats raise the stats as well as leisure riding encounters, but is there another way that raises stats quicker? I looked through the game guides and didn't find anything. Any help is super appreciated!

Yes, taking part in competitions (if you counted it in training, then sorry for repeating). If there are enough participants, your horse places well (in top 3) and luck is on your side, you may collect quite many specialty stats (my horses sometimes get 12+ specialty stats in a day (in 10 competitions)).

I usually feed specialty treats, train in their specialty, and then show in their specialty. I'll then leisure ride. Is that really all there is to it?

This is answered under Help > Showing > How do I increase my horse's stats?
What has been said pretty much covers it. Other ways are to feed peppermints or hay cubes, but not recommended for show horses. If you want to use it for show horses, then you need to use the converting discipline method, where you set a wrong discipline to cube in, then convert to the right discipline (and change to the right treat) later on.