You don't actually have to be registered in the organizations to show in the household pet class, you just have to fill out the show's entry form.
In TICA, if you want to get household pet titles you have to register your cats, but they can enter one show with their registration listed as "pending". This means if you wanted to try out a show before paying to register in TICA, you can enter the show and put "pending" where the entry form asks for the cat's registration number. After the show you can register your cat and claim the points from the show. In CFA they don't give titles or keep track of points with household pets at all. Some of the local cat clubs do give awards to the household pets and keep track of points for CFA shows though.
Show entry fees for household pets can vary. In CFA the household pet entries are usually cheaper, often $25-40. In TICA the entry fees are the same for purebred and household pets since they can all earn titles, and they tend to be around $50-70 for a two-day show. I've never shown in ACFA so I don't know what their fees are like.
The way I started was in a "fun" show. This was part of a big pet expo, and it was just a stand-alone show for household pets only, not part of any of the cat organizations. The judges explained a lot about how cat shows usually work and it was a little bit more informal than a regular cat show. I liked it so I looked up other cat shows coming up and entered a few. I found it helped to join a Yahoo Group about household pet showing to get more info on it as well.