Biking in and around Palm Springs is lovely in the mornings! Mostly everywhere you go will be about level so using a Beach Cruiser is fine. Palm Springs has wide sidewalks and many bike lanes in the streets. The beauty is unique since we are so close to the mountains and in the midst of all the Palm Trees. Riding in the neighborhoods closest to the mountains is downtown Palm Springs is just so relaxing. You can get bike maps from our local bike shop Palm Springs Cyclery near Ramon & Hwy 111 http://pscyclery.com/. The maps I used showed the length of the routes which was helpful for this amateur!
Be CAREFUL out there. This is a Retirement/Vacation community. Many people are unfamiliar with the roads and are not used to Bike Lanes even though the Bike lanes are fairly large. And some elderly drivers are not as sharp and aware as they used to be.
For Mountain bike riding Ive only spent time on the trails off Hwy 111 & Gene Autry in Palm Springs. There is a fire head trail on the left side of Vons market. Once you pedal for about 10 minutes up the steep fire trail there are numerous trails to ride up there. I have seen Specialized multiple times at this location test riding their full suspension bikes.