Don't do it in June. Or July. Or August. Or early September. You may not want to do it is February either

DSCN1893 by Stuart Black, on Flickr
DSCN1884 by Stuart Black, on Flickr

Taken on a very cold 22 February 2019 during a record snow fall.

But if you do want to do it in June do the Bicas Hottest Day of the Year Ride. But, generally, the loop around on the Santa Cruz/Rillito River trails with associated connectors is a pretty good ride. The whole "Loop" in Tucson includes a lot of other bikeways and connectors and is 130+ miles long so you might want to consider some short cuts in some places.

And, if you haven't ridden in Tucson, it is the land of pokey stuff. Saguaro aren't anything to worry about. The Teddy Bear cholla, on the other hand, will reach out and bite you

DSCN1875 by Stuart Black, on Flickr
DSCN1876 by Stuart Black, on Flickr

They make mountain biking down there an exercise in precision

image by Stuart Black, on Flickr
image by Stuart Black, on Flickr

Maybe try here: https://www.bikeforums.net/southwest/

did a little segment of it beginning of february. well signed and well paved with plenty of entry/exit points. hoping to get back there within the next year to hit mt. lemmon, ride the 9.5 mile loop
of saguaro national park-east portion again and do a bigger portion (30+ miles) of the loop.

the regional forums would seem to be the best place to post queries about riding places that you're traveling to but many of those particular regional forums
are dead as a doornail and not very helpful ime/imo. the southwest one is one such deadend-usually. there are a few of us in the southern california forum that are pretty helpful or try to be.
historically, i've had pretty good luck posting the occasional "hey...gonna be riding ____ for a few days. recommendations?" styled threads in the touring forum.
the ladies and gents that make up the touring forum have been around the block once or twice and are super helpful. you were right to post here. this forum is jumping.

That is a great picture of the Seguro cactus/desert. I was there once back in the mid 70's in the spring and saw the blooms. Beautiful! Was not bicycling.

Snow like that is rare in Tucson. The ranger at Saguaro National said that she hadn't seen anything like it in 30 years.

The mountain bike pictures were taken on Fantasy Island which is at the southern end of the main loop around town. It's actually a fun ride and not too hard. The route is about 7 miles long but, I swear, it doesn't go outside of a square mile area. It's a bit pokey but as long as you keep your elbows in and don't try to cut corners, you are fine

... Or March ....GloBULL warming LOL
Palm Springs hasn't hit 80F since middle of Nov. The record was 88 days, 3 weeks ago I think. It's going past 120 days likely.
Going back to the 40s for a couple days. My cousin stays there 3 months to ride MUPs. 3 inches of rain Valenine's day ate their golf courses and roads.
In 1997 I drove thru Tuscon, T+C and Tombstone, then off to California. Was there in April, was just right temps.

Are these mainly Mountain Bike trails? Are there plenty of good Road Bike tours?

perhaps you missed this spot. amitabha stupa and peace park.

For the cholla burrs carry a comb

I just cycled The Loop a few days ago. Just hop on anywhere, don't worry about maps. It's extremely well signed with excellent pavement. I don't think I went much more than five miles without seeing a water fountain and bathroom.

It wasn't easy finding the distance of the basic loop. Tucson brags that it's 130 miles long, but that includes spurs and both sides of waterways in many cases. It's actually just over 50 miles. Total elevation gain about 1000'.

I had no problems with the heat doing it in August a couple years back. Start at dawn with lots of fluids, be done by noon, enjoy a good long drink when you're done.

I did have a problem following the trail in NE Tucson. IIRC they had closed part of the trail to rebuild it (or de-build something else), and I took a signed bike route on the road, which then disappeared. Traffic was pretty well behaved, pavement was variable on the streets, and I found my way back to the trail after a few miles. Get the route map from one of the many bike shops in town and take it with you in case something similar happens to you.

If you can do climbs, head up the road to Mt. Lemon -- it's a lot cooler if you drive up to 4,000' and ride from there. Beautiful scenery, generally decent road and polite traffic, and coming back down is an absolute blast!

Ditto that suggestion. Mt Lemmon is probably the best road climb in the western US. It has a dedicated climbing lane all the way. And though you have to share the lane on the descent, it's so fast all the way, no traffic will need to pass. There's one stretch on a ridge it feels like you're in an airplane coming in for a landing. I generally do not enjoy descents that much.

And yes, the first 2000' in summer can feel like an oven if you have a tail wind. That can be rough.