English and hunt seat classes can be very similar to western classes. For an English pleasure class, judges look for a horse that looks nice to ride. You want to keep your horse at a consistent speed all of the way around the ring without speeding up, slowing down, or breaking, and keep them on a pretty loose rein. Make your horse nice and relaxed for pleasure. This class is based mostly off of your horse rather than your riding skill (for the most part, it takes a lot of skill to keep a horse moving easy like this).
Hunter under saddle, much like pleasure and equitation classes, ask for you to walk, trot, and canter in both directions. In these classes you may also be asked for a hand gallop, or to halt and stand on a loose rein. Focus on smooth transitions in hunter under saddle classes, and it is also helpful to use a horse that will back and stand fairly quietly.
In equitation classes, you are judged on your riding ability. You may be asked to walk, trot, and canter in both directions while keeping the perfect position in the saddle. Be sure to keep your heels down and your toes facing forward, your shoulders back and your back straight, do not slouch, keep you chin up, and your leg underneath you. If your horse acts up, the judge may look at how you respond and handle the situation.
Some shows may have a trail class, where you may be asked to do different tasks on horseback. You may have to open a mailbox and show the judge the mail, trot over cavaletti, do a serpentine, do a 360 in a box, back-up, L-shaped back-up, etc. You are judged on how well you perform each task.
You may have to also do command class. In these classes, there are multiple versions but the judges usually explain the rules first. Most of the time, the judges will say something like "All trot, all trot........NOW" and you cannot trot until they say now. If you break or are on the wrong diagonal, the judge will call you out and you will have to walk into the middle. When asked to canter, your horse has to pick up the correct lead or you may be called out. When asked to halt, your horse cannot move a hoof, unless they pick it up and put it back in the same spot. Focus on listening to directions, and staying calm.
I hope that these tips helped! I'm not sure that you will experience all of these classes, but you probably will compete in some. Good luck!