The National Parks are a great place to start. You are very close to Zion and Bryce. We just went to Zion and the Grand Canyon. Zion's Watchman campground has sights right next to the river and has electiracal hook-ups. We also like to go to Yosemite. The parks have lots of hiking and biking, plenty of beautiful scenery along with Junior Ranger programs for kids. These campgrounds are also inexspenive but require advance reservations. nps.gov can give you more info.
State parks are also great and cheap. We like to camp at the beach a lot. San Clemente is our favorite. Doheney, Carlsbad (near Legoland) and Malibu Creek are good too.
reserveamerica.com is the online reservation system for state or national parks. I don't remember which at the moment.
I've always wanted to camp at Red Rock near you. Have you done that yet? We stayed at Oasis in Vegas and weren't that impressed.
How far are you willing to drive? I grew up camping at Lake Silverwood in the Cajon Pass. Calico Ghostown is a fun place. Down south in Sand Diego there are great camping places such as Campland on the Bay and Santee Lakes. Santa Barba and the Solvang area have several campgrounds.
woodalls.com is a wonderful resource for campgrounds across the country. Check out Fogorn books Nevada Camping or California camping. We've found lots of good camping through our Ca book. Both are available on amazon.
We've had our trailer 5 years and have had so much fun. Your family has lots of good times ahead!