...Btw, my goal isn't to necessarily compete. I want to get to a point where I am capable and comfortable taking a horse out to ride on my own. Should I even care if the instructor specializes in dressage or jumping? If so, which would you recommend for someone hoping to one day be able to own her own horse, and take it out for trail rides in the country?
I would not consider either to be a qualification or disqualification. Lots of folks who teach jumping or dressage will also teach general purpose riding. However, some also prefer to focus on one or the other and may not want to teach toward a goal of trail riding. It depends on their business model.
The lady who taught my daughter came from a barrel racing background, although she does not specialize in teaching barrel racing. She's about 80% horse trainer and teaches riding on the side.
But she did something I really liked: when meeting to discuss taking lessons, she asked my daughter in detail about her goals. She asked questions like, "If you could describe heaven being on a horse, what would you and the horse be doing?" As they talked, she realized my daughter's goals were to be able to go riding safely on Trooper - the only horse my daughter has any interest in riding. She didn't want to compete in anything, she just wanted to be able to ride him safely in the local area and have a happy horse.
The lady took that info and taught her toward THAT goal. Lessons focused on things like keeping your balance when something strange happens, or how to read your horse so you know when he is happy and when he is uncomfortable. The lessons included how to calm a horse down, and how to behave so he would enjoy being ridden by you. She discussed how to get a horse 'softer' - to be more sensitive to his rider. From time to time, she'd check to see if my daughter's goals had changed. They didn't.
After about 6 months, she told her she was probably safe to ride Trooper in the local area. If she wanted to go beyond that, give her a call. Otherwise, enjoy her horse!
My daughter enjoys Trooper. He adores her. She will sometimes ride another horse, but she considers riding any other horse work instead of fun.
If you talk to an instructor, and the instructor doesn't seem interested in YOUR
goals, then find someone else. They should be there to help you, not the other way around...