Been there done that... several times. At Zion make reservations {as early as possible} for the Watchman campground which is located inside the park and short walk to the visitors center/tram station. You get 30 amp service for $18 a night {$9 with a Geezer card}. They also have a two lane dump station. Some of the sites are right on the Virgin River... Awesome!
Be advised that Zion in June and July will be two things... HOT and crowded. You can mitigate the heat by getting out early and hiking to the higher elevations but expect triple digit temps every day. The water in the Virgin River was 48 degrees in June so you can beat the heat but brrrr. :S
As far as Canyon Lands and Arches which are right up the road from Moab, they too will be hot and crowded {In April we waited 45 minutes just to get into Arches as the line of vehicles stretched all the way back to the highway - Trust me June will be worse so get there early}.
Don't miss Capitol Reef NP on your way to or from Moab as it is one of the most spectacular NP's though lesser known. Kodachrome Basin and of course Bryce Canyon are two other awesome areas to explore.
:C