First of all, I recommend not thinking of dressage as a sport. Especially if you have no plans to go deep enough into it to become competitive. What it sounds like you are interested in doing is learning some of the fundamentals so that you can better help your horse be balanced and work correctly to improve their performance in the sport you ARE competing in, if that be jumping .
I think the first thing to work on is your seat. working on lengthening the leg and having a well aligned body, no matter what gait the horse is moving in.
Then, you start working up the training pyramid. The first level is Rythm and relaxation. Getting the horse to move forward smoothly, with a rythmic gait , relaxed and using a good amount of push from behind, but staying loose in the back.
That's enough to start with.
Also, keep in mind that if your horse has never done anything like dressage, they may not really enjoy being your school horse for learning dressage. some hroses really hate doing dressage. others do well at it. depends on the horse. It's no fun if you are trying to learn on a hrose that just hates this kind of slow work, but would rather dash around barrels, or jump a course at high speed.