The good and bad news is that there are so many places to photograph that it could fill a lifetime---one of the reasons I am considering returning . . .
Tahquitz Canyon---There is a good place but it is on a reservation. Using a tripod is questionable, but there is a nice waterfall a then a HUGE palm grove if you are up for the hike. Its going to start getting warm soon. April is naked weather.
The boulders up the Tramway might be good---but there might be snow . . . The Palm to Pines Hwy that goes South out of Palm desert is a great road. Sometimes there are hangliders. Anza-Boreggo can be cool. There is the Salton Sea to the East which can be cool too.
To the North-East is the 1000 Palms Oasis and Preserve. A one mile hile takes you to a big dune (that you can't walk on) and a big pond that reflects the palms---another favorite place---if for nothing else than a place to stop and eat a grilled cheese from the In and Out 5 miles south.
Morongo Canyon Preserve (near Covington Park)is nice. It is north on 62 about 20-30 minutes (fkn traffic). there was a bunch of rain so the marsh there should be full of stuff. Pioneertown is pretty cool---beautiful is more like it (go north at the WaterCanyon in Yucca---they have good sandwhiches and half-decent coffee) there is a nature preserve up there that I passed a few times but never went, I was looking for the back road to Big Bear.
Josh is the best bet though. 45min into town(get a burrito @ Santana's, and they are super nice at Coyote Corner) and another 10 to 15 up to the "good stuff" in the park. Key's view is great on a clear day, and makes you reconsider fossil fuels on a not-so-clear day (of course, my favorite view is only 3/4 of a mile from the west entrance---it is a secret though). One of the best places in the Monument is the Indian Cove campground and Rattlesnake Canyon (don't let the name fool you, I was never bittten there) there is a dried up waterfall to the South-East of the area---nice pools of water sometimes form. Blackrock is nice too. (A GREAT drive is to go up to Eureka Peak---turn north on La Continta Road and continue north when it turns to dirt---stay left or it will turn into a dirt bike trial (much prettier, but a ruined a car that way). There was a fire a number of years ago but I imagine it is quite nice now (it was nice after the fire too, just a little)
Death Valley is nice if you've never been there. but it really is a LONG way and you end up spending most of your trip in the car.
I am now starting a series of photographic guide books of the Morongo Basin---