In June, Zion will be a circus with traffic. I would suggest Cedar Breaks followed by Bryce Canyon then head over to Canyonlands via scenic back roads. You could also head to Cedar Breaks via a short stay in Ely, NV then head across to Cedar City stopping along the way to see Lehman Caves in Great Basin Nat. Park. You would miss some of the 110 degree heat that route as you gain 4,000 feet in elevation after LV.