Joshua Tree is mostly a place to drive the road from the south to north entrance or vice versa and get out to look at rocks, trees and views.
Palm Springs and the adjacent towns have several nice (although fairly modern) downtowns to wander. Good hotels, restaurants and shopping. You can also go up the mountain on the cable car.
For a two day trip I’d suggest one day driving through the National Park (maybe the second, then you just arrive back in Phoenix late evening) and one more leisurely afternoon after arrival relaxing in Palm Springs/Palm Desert.