I'm in Riverside Peter, just an hour from PS..
I second Joshua Tree as one can spend an eternity there and find wonderful images. I was just there last Sunday (with Per) to catch the snow..yes, snow. JT is at around 4000 feet plus...and the next storm is brewing as I type.
Not sure when you are coming, but if soon, you may get lucky with the white stuff...quite pretty in the desert.
The wildflowers will be blooming there in April, depending on conditions.
Other than that, you can take the Palm Springs tram up to Idylwild (Mountains), though there will be snow up there for a bit as well..
An hour and a half away are the San Gorgonio mountains (Lake Arrowhead & Big Bear Lake), and Death Valley is around 4 hours, maybe five...long day trip...
The pass (60 FWY west) between Palm Springs and Riverside (Banning/Beumont area) has some nice scenery and back roads...or, the Oak Glenn area (apple orchards) should you take the more northerly 10 FWY heading west.
Oh, and all the girls in bikinis on spring break in downtown PS are always worth a look or stare, if you're timing works..
Matt's Photo Site
"I invent nothing, I rediscover". Auguste Rodin