LAX is far from Palm Springs but it would be less expensive to fly to LAX. Then you would have to rent a car or get on a busor tour to Palm Springs. There is also the Palm Springs Airport but it would be pricey to fly into here. The Ontario Airport is another airport you could fly into.
There are limited flights into Palm Springs and you'll definitely pay extra for those. If you want to save some money and don't mind a little drive try flying into Ontario, CA. You can rent a car and drive out east on Hwy 10 to Palm Springs (about an hour). During the drive you can view the desert and get a true appreciation of the oasis that Palm Springs is in the middle of a sandy desert. You can also pass a casino and outlet mall that are in Cabazon (20min outside Palm Springs).
Palm Springs isn't that big. Some of the planes that fly here are pretty tiny. When we arrived in Palm Springs from Los Angeles, they had to rearrange the passengers in the 29-seat plane and they left our luggage behind as there was not enough room for it.
We took the same 29-seat propeller plane when we left Palm Springs three weeks later.
However the nice thing about a small city is that everything is just minutes away. Downtown is 10 minutes from the airport. My brother's house is just 3 minutes from the airport - or a 15 minute walk.
Fly directly into Palm Springs or fly to Ontario California (typically cheaper) and rent a car. The drive is about an hour.
Car or bike. If you stay close to downtown you can pretty much walk or bike everywhere you want to go. If you plan on getting into the countryside you will need a car.