You might not think of Las Vegas as an outdoors town, but Nevada is more than just desert and slot machines. Las Vegas is actually the hub city for a large area of public land that many forget about.
The best camping near Las Vegas (and the most well-known) is at the Lake Mead National Recreation Area. Lake Mead is quite the popular spot for both RV and tent camping. If you want to day trip to Lake Mead and find free campgrounds nearby, you have BLM land in almost every direction. If you want more traditional accommodations, you have Boulder City or Henderson, NV, just nearby. On the other side of Las Vegas is the Spring Mountains National Recreation Area, with more than 100 campsites available.