if staying in palm spring/palm desert area, you'll definitely want to be off the roads by 10am with the heat that time of year.

the closer you are to the 10 freeway, the windier it will be-especially on the western end of the valley by the windmill farms.

best close climbs are tram road in palm springs, the shortish climb at the end of frank sinatra rd in rancho mirage, highway 74 up to the art smith hiking trailhead parking (just before the road gets really wiggily) and 1000 palms rd across the 10 freeway (accessible from ramon rd or washington rd).

favorite noodling around areas are the movie colony area in palm springs (east of indian canyon-north of alejo, south of tachevah, west of avenida caballeros) and the vista/old las palmas area (south of the tram, west of palm canyon dr aka hwy 111 and north of the palm springs desert museum. tucked up against "the mountain" and adjacent to the movie colony, there are oodles of spanish-style and palm springs modern houses. the closer to the tram, the hillier it is. also like the area at the southern end of palm canyon dr towards the indian canyons which is a shortish ride to/from the movie colony and vista/old las palmas areas. if you find yourself closer to palm desert, i like the la quinta cove area for an easy spin.

joshua tree national park has been mentioned and either the west or south entrance from palm springs/palm desert is the best part of an hour way by car. a little cooler but still gonna be warm in may. parking just inside the west entrance of jtnp 5 miles from the town of joshua tree and gently climbing up to the hidden valley area 15-20 miles round trip)will be your best bang for the buck but you can hit the southern (cottonwood) entrance as well. riding from the town of mecca (at the eastern end of coachella valley) and heading up the gentle & scenic grade of box canyon rd, then crossing over the 10 freeway up and into joshua tree national park up to the visitor center is a worthy 50 mile round trip ride with a little over 3k in elevation gain.

if you want some alpine adventures, you can drive about an hour west on the 10 freeway to ca hwy 38 and climb up to forest falls, angelus oaks and onyx summit at 9k feet in the san bernardino mtns. long but awesome.
you can also take ca hwy 74 from palm desert up to the intersection of hwys 371/74 (30 min drive from palm desert/50 mins from palm springs), park at the paradise valley cafe, and ride west along hwy 74 up to hwy 243 and make a left up to idyllwild and return. about a 35 mile ride rt with approx 2,500 ft of elevation gain. for bonus climbing, you can hit the dead-end and low traffic roads of apple canyon, morris ranch and fobes saddle.