Sleep in the developed loop at Usery Mountain Regional Park, a few minutes down the road from the trailhead, and take Wind Cave in the morning while the south face of Pass Mountain is still in shade. The climb runs a mile and a half to an alcove partway up the cliff, exposed almost the whole way. By midday the slope bakes and the loop is where you want to be.
Book that night before you book the van. Sites open six months out on the county system, the developed loop fills October through April, and everything else about the weekend follows from which pad you get. If a van feels tight for the family, RV rentals in Mesa fit the same pads.
If the loop is booked, the next public desert campground east of town is Lost Dutchman, right under the Superstition wall, and that one is really a Phoenix trip. The Arizona campervan hub covers the rest of the state.
Saguaro Lake and the Lower Salt River, out on Bush Highway
Bush Highway climbs out of east Mesa to Saguaro Lake, a reservoir walled in by canyon rock with saguaros on the slopes above the shore. It's the water day people take on a weekend, tubing or kayaking or idling a boat under the canyon walls. From Mesa it's about 27 miles north on SR-87 to Bush Highway, then a few miles in to the marina, and the Lower Salt River comes off the same road where the tubing crowds put in.
Both are day-use for a van, so you run the water from the Usery night and drive back. You'll want a Tonto Pass for the lot, and the lake's only overnight is boat-access Bagley Flat at the east end. Summer weekends fill the lots early, so start in the morning.