In 2016 we visited the 5 National Parks and started our 'tour' in mid May. Weather was fantastic and the canyons were gorgeous. Even had snow at Zion and Brice. Highly recommend doing all 5, but take your time. If you can camp at Kodachrome State Park (near Brice Canyon), you'll love it.
We had a schedule to make (we live in Florida and had obligations back home) and I wish we had stayed another 4 days at each park: so much to see.
I live in Utah and I would suggest May. You still might get some snowy weather in the May as well as September, but the days will be longer in May and it will be warmer. In May the sun rises around 6:30AM and sets around 8:30PM. In September the sun rises around 7:00AM and sets around 7:30PM. Sometimes that extra hour or so of daylight can make a difference.
But, another thing to consider if you are traveling to the Moab area to visit Canyonlands and Arches National Parks, May is the busiest month of the year and there will be big crowds, especially in Arches.
2019 Rockwood 2109s
2016 Ram 1500 Big Horn Hemi