Joshua Tree NP is in the desert, so it's only natural that cacti grow in some places. Most of them are visible near the trees themselves. Though if you were to venture down towards Cottonwood in the South, you would come across a grove of Cholla Cactus.
One of the main attractions of Joshua Tree NP are it's extensive rock climbing opportunities. I'm not a rock climber myself, but it was fun to watch people do it. And just because you aren't a rock climber, doesn't mean you can't enjoy the rock formations themselves.