You don't have to be an ARBA member to show at ARBA sanctioned shows. If it is a 4-H show, then you would need to be a 4-H member. Youth shows with ARBA just require the person showing the rabbit to be between 5 and 18 years old.
To enter a show, you need to find one. The ARBA website lists all sanctioned shows, so you should be able to find one. Many shows will have a catalogue with info on when and where the show is, the entry form, and other relevant info. Some shows require pre-entry, so you have to send in the entry form before the show. Others have you enter on the day of the show. Entry forms are fairly simple, just need your info (each person should have their own form for their own rabbit(s)), and the rabbits tattoo, breed, sex, age, and variety. Some shows will have 2 or 3 shows a day (each would have a different judge or judges), so you can get more opinions on the rabbit in a day.
The actual showing is fairly simple. Just get the rabbit up to the table and put it in the right coop (ask if needed). The judge will do all the work, you just need to step back and watch. Once it is done, you can take your rabbit back to the carrier or cage. If they have won Best of Breed, they will come back for Best in Show. If your kids are showing in the youth show, they need to bring the rabbit to the table them self as a youth needs to carry the rabbit to the table (helps keep adults from showing in youth).
You do want to arrive on time for the show, even get there early if you can. Some shows let you set up the night before. Getting there early gives you time to set up your rabbits, make sure you are signed in and paid the fees (and enter if needed), and just get settled before judging. Since each show has a different judging order, you don't know when your breed will be up until you get there, they could be first or last.