Having done that trip several times 6 days really isn't enough time to do everything that you want so I would pick the route that has the one or two things you just have to see. That's a 22 hr drive if you do it straight, so just stopping in Moab, Sedona or the Grand Canyon you could spend a day in each of those places. Remember as well in June once your in southern Utah it will be in the 100's all the way to Phx
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