These days it's easy to cross the border into the West Bank. I highly recommend spending time exploring the other side of the wall.
The Church of the Nativity is believed to be located over the cave that marks the birthplace of Jesus of Nazareth. Even if you're not religious, it's deeply moving to experience this basilica. Make sure you don't miss the peaceful courtyard behind it.
Abed Yossef (firstname.lastname@example.org) is based in Jerusalem and has been my trusted driver and guide for the past years. Depending on sites and cities you want to visit in the West Bank, he will charge around $150–$250.