Seems like you already are aware of the nature choices;--the tramway will take you to the top of the mountain and the main hiking trails are all to the south and to the southwest of where you are staying.
If you go up the tramway at night, dress for cold weather. Other than that, you should be enjoying (as it appears you are) the nice 80 degree warmth of Palm Springs during this normally winter period. Palm Springs and Phoenix (Arizona) is where a lot of "snowbirds" go every winter from the colder climate areas of the US (and apparently UK, as well) to escape and meet like-minded transplants.
Have a great time enjoying the rest of your trip and I hope you return to the US again soon.
[ Edit: Edited on 23-Feb-2014, at 07:27 by Calcruzer ]