I've been to both and would definitely recommend Arches National Park (it's not called "National Arches", btw) for hiking. It is absolutely gorgeous, and there are lots of trails of different difficulty levels. Once you get about a mile from any parking lot, you'll encounter many fewer tourists on the trails.
Petrified Forest is mostly a driving thing, though there are some short hiking trails. It's fascinating too, but not nearly as much to see, and it's mostly just flat, open land. If you can only go to one, Arches is the way to go. Also, if staying in the Moab area, Canyonlands National Park is also nearby and is stunning, with many amazing hikes. I highly recommend it.
We did a blog when we visited many southwest parks in 2006. I can't believe it's been 10 years! If you are interested, they are here:
Petrified Forest NP: http://travelingtwosome.weebly.com/sw-parks-page-7.html
Arches NP: http://travelingtwosome.weebly.com/sw-parks-pg-11.html
Canyonlands NP: http://travelingtwosome.weebly.com/sw-parks-pg-12.html
I didn't know much about blogging way back then and did it as one LONG report with no easy way to negotiate between pages, but the content of the individual pages may still be helpful to you.
P.S. We did not do any mountain biking back then, but my former manager used to go to Moab every year and raved about the MTBing. He was a much more advanced mountain biker than I, however.