The Tram is a good starter, but honestly I found the roads in general there to be not very friendly for cycling. I’d recommend renting a car and riding in the Joshua Tree park itself. Lots of people are driving there, but it’s all slow and for whatever reason everyone was careful both times I rode there.
Another good option was climbing up to Idlewild on 243 from Banning. Beautiful route, but still that can be sketchy in the beginning because cars drive like it’s a highway. I did a route like the one on RWGPS that goes around the town, and frankly it’s not very pleasant given car volume, wind, etc.
I’d actually suggest using a gravel bike instead and exploring the mountains on the north east side of town. PSP itself doesn’t have great mountain access.