There are dozens of hiking trails all around Palm Springs, most of them just a few minutes away from town.
A good moderate hike with wonderful views is Tahquitz Canyon in Palm Springs. The entry is at the end of Mesquite towards the mountain off of S. Palm Canyon. There is an admission fee of $12.50 adults/$6 children under 12, and there are guides if you like. You can also watch a short film on the Legend of Tahquitz, a Cahuilla Indian shaman who is said to be seen as a ball of green light.
Another great place to hike and explore is Indian Canyons in south Palm Springs. There is an admission fee of $9/$5. One of my favorite hikes is the Murray Canyon Trail just after you enter the admission gate. It starts off flat and open, then winds through a beautiful canyon to the “seven sisters” falls which run in the spring and early summer. If you take the main road all the way back, there is a trading post and many hikes through an oasis and along the creek.
Good (and free) local hikes can be found in the book “140 Hikes in and near Palm Springs” by Philip Ferranti. Most popular are the various segments of the Lykken Trail, the Skyline Trail, Oswit Canyon, and the many trails available at the top of the Palm Springs Tram.
Recommendation: Hiking in Palm Springs can be deceptively dangerous in warmer weather. If you are planning to hike in the warmer months, read this safety guide before you go: Hiking Safely in Palm Springs