Tons and tons of great riding in the Moab area. It will likely be pretty hot in June. Get up super early... hit the trail, then siesta during the hot part of the day... then enjoy the late afternoon/evening colors and temps.
Also if you care to escape the heat just head up to the La Sal Mountains or deep into the canyons to find shade... Or just head up the hill into Colorado.
The Castleton-Gateway road through Little John Brown Canyon is a nice short cut into some incredible riding between Grand Junction and Telluride and you can find passes that are miles high if you just head further into the big Rockies.
There is so much to explore in a 150 mile radius from Moab it'll make yer head spin! Have a blast. Hit me up if you want some recs. CubDriver is down there now with a list of hotspots I gave him. I'm sure he'll share upon return from this recon run. Otherwise I'd be happy to share.
There are also the two new plotted and mapped Backcountry Discovery Routes too. One in Utah and one in Colorado. Butler maps is a great resource for excellent motorcycle roads and trail routes.