Hunters consists of hunters over fences, equitation over fences, and medal rounds.
Hunters is scored on the horse. You want a horse that takes every step the same size, it's not speeding up and slowing down a lot throughout the course. It's consistent.
Equitation is judged on the riders posture, but also their ability to execute technical maneuvers, such as rollbacks, and bending lines. The pace should also be consistent, but it's not the key factor. You want to see a rider riding every step of the way.
There's also medal rounds which are based on the horse and rider. These can be judged different ways with different weights of scoring on the horse and rider.
Another equitation round worth looking up is the collegiate riding IHSA national championships. This is equitation riding, but competitors are given a random horse from a drawing to ride. This will even out the playing field so that it's judged on the rider's ability, and not how fat their wallet is.
There are different jumping rounds too, such as gamblers choice, six bar, and puissance.
In gambler's choice a course is set with several jumps, each set with different point values relative to their difficulty and height. You com up with your own course to collect as many points as possible within a set time. You can jump most fences twice, and you can jump some both directions.
In six bar, there are 6 fences set in a series, usually a few strides apart. After the clear round the fences are raised higher. If your horse knocks a pole they are eliminated. The horse who makes it through the highest wins.
I haven't ever seen puissance in person, but it's a single fence that is continually raised, similar to the six bar.
I think it would be worth while to look into the different types of jumping classes and see the horses that ride in each type of class. The horses and rides can be quite different from one another depending on the class!
Growing up I saw a lot of horses come off the grand prix circuit who went on to be really successful in the hunter ring. Many of the riders went the opposite way, starting off as hunters and becoming jumper riders.
If you ever have a chance, check out the HITS show series. The best riders from your area will be there!