Rabbit shows would be similar to cat shows in that the judge does the posing and evaluating. I don't really know about rat shows, so can't comment on them. Dog shows have the handler posing and running with the dog. For the actual showing, you just need to get your rabbit to the table at the right time and then stand back and watch.
The rabbits do have to be purebred (or at least look purebred) and be intact. A pedigree is not required unless you want to register you rabbit and registration is not required to show. Each breed has it's own standard and accepted colours.
Rabbits are shown in classes that are based on breed, age, sex and colour variety. The colour varieties can vary from breed to breed, some it is just solid or broken pattern and others it is the actual colour. Age is junior, intermediate and senior. Junior is under 6 months and senior is over 6 months, intermediate is for the giant breeds and is 6-8 months. Breed is the breed and sex is buck or doe. If a rabbit wins their class, they move up to Best of Breed and eventually Best in Show. Some shows will do Best 4 and 6 class or even divide the breeds up into groups.
Each show is a bit different. Many require the rabbits to stay in their carrier for the show, but others provide cages. You should bring your own food and water. It is good to clean out the carrier as well, probably at least daily.
Each show should have a show catalogue and entry form. The catalogue will have the info on the show from where and when it is to the judges and specific rules for that show. The entry form is just what you need to enter. Some shows require you to enter in advance while others just do entries on the day of the show. Payment can usually be done at the show.
Basically what you need to do is check the catalogue for the rules and get your entry in on time. Show up early so you can pay, check in and get set up. Some show do set up the night before and you can leave your rabbit overnight. There should be a list of the breed judging order, so that should give you an idea on when your rabbit will be up. At the right time, they will announce when each breed it us, get your rabbit to the table. There will be small coops to put them in and there might be one for him, so make sure he gets in the right one. Then step back and watch and listen to the judge. When it is over, take your rabbit back to the carrier. If he got Best of Breed, he will come back later for Best in Show. You usually do have to stay for the whole day, so bring something to do. There are usually auctions, raffles and other things going on as well to help raise money for the host club.
If you can, it can be a good idea to go to a show to see what it is like. Don't enter your rabbit and just watch. You don't need to be there all day, so a few hours should be good to see what goes on.