I would also highly recommend Dead Horse Point State Park. The area is truly awesome and would be ideal for bike riding. It's close enough to do things in Moab.
I'd also recommend staying in Arches National Park if you can do without hookups. You'll be right there for tremendous hiking. Regardless, sign up in advance for the Fiery Furnace ranger-led hike. You'll never do anything like it again. It's very popular so reservations are a must. Don't wait until you get there.
Early September can still be hot for hiking and biking. Later through October would be best.