Usery Mountain Regional Park keeps the closest campground to Gilbert, 73 developed county sites out where the Valley's east edge runs into Tonto National Forest, with water and electric and room for a rig up to 45 feet. The cool months fill it, so book a site before you leave home. Usery is really Mesa's park, and the Mesa campervan lander covers the Wind Cave hike and the rest; from Gilbert it's the honest first night out.
If it's full on a busy weekend, the next campground east is Lost Dutchman, under the Superstition wall standing straight off the desert floor, and that overnight belongs to the Phoenix lander.
San Tan Mountain, the day hike south of town
San Tan Mountain Regional Park runs south of Gilbert past Queen Creek, more than 10,000 acres of creosote flats rising into saguaro along the escarpment. It's day-use only with no camping, so you hike it in the morning and drive back to Usery to sleep.
If a campervan feels tight for the group, you can size up to an Arizona RV rental and still fit the same developed sites and the same East Valley loop.