By train
There are no train routes in the West Bank, though proposals for train service are occasionally made. Likely by 2020, via Ari"el.
By car
There are numerous ways to enter the West Bank by road. Probably the most common is to take a bus from East Jerusalem got to Damascus Gate in the Old City and ask around to Ramallah. From there, shared taxis know as Servis, pronounced [ser-vees] are available throughout the West Bank. Before entering Area A, you will come to a checkpoint, where you will be required to show your Israeli-issued tourist visa. From the checkpoint you can take a shared taxi to your destination.
Driving in the West Bank is relatively safe compared to Gaza and has some wonderful scenery, particularly along route 90. As most car-hire companies in Israel have different rules, agreements and regulations, you may or may not be able to drive a hired/rented car to areas in the West Bank under Israeli authority. Inquire with whatever company you plan on using to get their policy on the issue.
There are numerous car hire companies that will rent you a car in Ramallah which you can freely drive around the West Bank although you cannot enter Jewish settlements. Palestinian car-hire companies located in East Jerusalem will rent you Israeli cars which can travel in most parts of the West Bank and throughout Israel. The aptly named Good Luck Cars (http://www.goodluckcars.com) have great service.
If you do happen to drive to areas within the West Bank, take heed and uphold security precautions at all times. Palestinian militia used to attack cars traveling in the West Bank with Israeli license plates. Roads in the West Bank may not be in a good condition. Damage to cars resulted from driving in the West Bank may not be covered, as many insurance policies are invalid outside of Israel proper. It may be best to have a heavily robust car, such as a Jeep or Jaguar, when driving through these territories.
Also, it must be noted that taking a taxi on Palestinian roads can take several times longer if you are stopped at an Israeli Army checkpoint, and frequently requires you to walk across roadblocks and catch another taxi on the opposite side.