Dang DaveNW, that is a seriously ambitious trip y'all are putting together but you picked some of the most stunningly photographic areas in the Four Corners. Have you looked into flying into SLC and then doing a puddle jumper to Durango or Farmington for car rental?
What are the most desired areas for you to visit? How much time do you want to spend in each area? You could spend a week in just Canyonlands and Arches and never want to leave.
Getting to the SE corner of Utah is not that easy. Used to live in Monticello and playing in Canyonlands and Arches was a way of life. Late spring will be lovely. Even the NEAZ stuff will still be comfortable weather for you.
If it's more economical flying through a major hub, then hitting Phx first and going north is how I would do it. No matter which direction you are still doing a LOT of driving. But it's easier getting to NE AZ from Phoenix than it is to SE UT from SLC. Have made the SLC to Moab trip more times than I want to admit and, while admittedly biased, it's not my favorite area of Utah to visit. The LaSal Mtn area is "meh" compared to the stunning beauty of the Uintah Primative farther north.
Not sure how you are thinking of looping in Antelope Canyon (so worth the visit though). Easiest access is through Page. Traversing the Dine' Reservation area has gotten a lot more friendly in the last 15 years but the road infrastuctures are still a challenge. Tuba City and Chinle have grown up a lot but getting to Page from Chinle you will for sure have to backtrack, not loop.
Did't stick to your original question of car rental out of Phoenix. Nertz.
. Got distracted with the trip itself. Have been and played in all the locations you mentioned. Before you decide on rentals check the tax rates for car rentals out of both airports (SLC / PHX). Both are high. My DD always rents in SLC from non-airport locations in downtown SL when she goes north. Renting off-site in Phx is always cheaper, but Tempe seems a little far. There rental sites closer in downtown Phx. Might be easier to rent from a location where your friends live if they are picking you up though.