My family camps often and my children are now 4 and 7. We also live in Long Beach and prefer to not drive too far. Some of these locations are a little farther than 2 hours, but worth it. Based on what you're looking for I would look at El Capitan State Beach, Carpenteria, Lake Casistas, and San Mateo. You can reserve these on reserveamerica.com. An expanded list I wrote to another mom a couple of months ago is below. Have fun!!!
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We, too, love the beach. We favor beaches in Ventura County...El Capitan State Beach, Carpenteria, Faria (county park), Thornhill Broome (part of Point Mugu), Jalama Beach (county park). We have friends who like Refugio, McGrath, Leo Carillo, San Clemente, San Elijo (in San Diego County). We love desert camping at Red Rock Canyon. A fun place for boys to climb!!! We also enjoy Barton Flats, Dogwood, Cuyamaca, Big Sycamore Canyon, Mammoth (mostly Convict Lake and Coldwater), and more. Have heard great things about Lake Casistas and are planning a trip there soon. There are more, but I can't think of all the names right now. Regardless, I think I gave you a good start. Also, a lot of people like Doheny. It's very popular. The campground is nice, but I have issues because the water is always contaminated and we like to play in the water. San Mateo is nice, but not a favorite. Anyway, feel free to email me for more info. And, HAVE FUN!!!