I've never been to the Petrified Forest, so my enthusiastic endorsement for Arches NP (and nearby Canyonlands NP and Dead Horse Point SP) is very biased. Moab is known for its many great mountain biking options (from beginner to advanced), but there is also beautiful road biking to be done. One MTB trail I can recommend is the Klondike Bluffs trail - it's doable for a beginner/intermediate, and it will take you to a gorgeous overlook of Arches NP.

I don't know what time of year you are planning to go, but I recommend spring or fall. It gets extremely hot in the summer.