A few answers and comments...
Shows are classified into different categories, technically the distinction is the amount of prize money given. To be classified as an "A" show you have to give out a certain amount and to be an "AA" show you have to give even more (and these are governed by USEF). Then there are "C" shows which are local level or "schooling" shows. Schooling shows are much smaller, much more casual, and an excellent place to learn the ropes and/or bring green horses. They may be more forgiving of mistakes, the competition is usually not horribly competitive, it's often ok not to braid or wear tall boots as opposed to half chaps. Things like that. At A shows many of the riders are from out of state, the competition can be challenging, and you really will stick out if you aren't braided and dressed traditionally.
My first suggestion is what other people have said, find a good trainer! Second, would be to start off at a schooling show. Your horse is pretty cute! I also agree that she could do well at lower level hunter classes. What is meant by that is a class that's not very high (therefore not super competitive) at a show that is not super competitive. To show in the 3" divisions at A shows your horse MUST have a lead change and the courses will be set on at least a 12 ft stride. She must be a very good mover. And while your horse is really quite cute she isn't quite up to par with some of the fancier warmbloods you see in the upper level classes these days. Some local circuits are pretty challenging even if they aren't A rated. Our local circuit offers about 10 shows a year and a lot of those horses go to a few of the A shows around here so it's not easy. It's a well run circuit with top notch judges (well, most of the time!) even though it's casual enough that we don't braid manes!
The main difference between hunters and jumpers is what they're being judged on. Do a search on here, the topic has been covered a bazillion times but what people have said is right, hunters need to be quiet, consistent, get every distance correctly, jump round, move with a long flat kneed stride, etc. The highest they are offered is actually about 4" in the Regular Working Hunter division, but I recently saw a 4"3 performance hunter division. Your horse moves decently and has a cute jump. Cute expression overall. At our local shows she must have a lead change to be in the ribbons. Other circuits allow for a simple change. That's the beauty of the schooling show.
Equitation is actually a type of class that is not based on the horse at all, but based on the rider. They are usually flat classes AND jumping classes. They will be judged not only on your position but how well you can navigate the course and how well you ride on the flat.
the Texas Rose Horse Park is an EXCELLENT and beautiful venue I love going to! The prizelist looks like it offers some great classes to get your feet wet.
Hope that helps a bit!