Travelling around Israel and Palestine

There is public transport in the West Bank, but it isn't really my area of expertise. Buses leave from outside the Damascus Gate of Jerusalem's Old City going to Bethlehem, Ramallah etc. You won't find any online timetables for them.
You won't be stuck for hours at checkpoints. As a tourist you will almost certainly be waved through. You must keep the paper visa you get at the airport because that is proof of your tourist status.
For buses to the Dead Sea and Masada, see www.egged.co.il/eng.
Jerusalem to Tel Aviv and back, by bus. There are two routes from Jerusalem to Tel Aviv, both leaving from the Jerusalem central bus station. Route 405 goes to the Tel Aviv central bus station, and route 480 goes to a terminal next to Tel Aviv Merkaz Savidor railway station in the northern part of the city. Both routes go every 10-15 minutes and take an hour. The fare is 19 shekels one way. Buy your ticket on the bus.
Note that in Israel there is no public transport from about an hour before sunset on Friday (meaning last buses reach their destination at that time) until after dark on Saturday. Shared taxis between Jerusalem and Tel Aviv run 24/7, leaving when they are full.
West Bank buses run every day.

There are shared taxis and minibuses in the West Bank. I don't recall exactly, but I think they are unmarked. In larger towns a good spot to catch one is (are) the road(s) at the entrance or exit of the location. In small villages they stop at whatever instinctively goes as the "centre" of the settlement - the mosque, municipal office, biggest shop or whatever. Stops are not marked. Ask around or look for travellers hanging around by the side of the road. You can always get a ride to the next village or to the nearest bigger town. Direct trips from one small place to another small place far away are not usually possible. Overall, the whole system is informal, friendly and I think I never waited long for a ride.

All in all, traveling around the West Bank is fairly straightforward. You should always keep your passport on you, but don't expect much hassle or time spent at checkpoints. The exception to this is at Qalandia Checkpoint going from Ramallah to Jerusalem. It's always reminded me of a Prison, both in appearances and the way the guards treat people. If you cross from Israel to the West Bank and back frequently, expect the level of hassle you receive when returning to Israel/Jerusalem to increase.

You can find public buses to most cities in the West Bank from one of the two bus lots near Damascus Gate in Jerusalem. Ask around locally or at one of the bus stations and someone will point you in the right direction. For smaller destinations and villages within the West Bank there's a taxi stand a few blocks north of Al Manara (the central square) in Ramallah. Again, ask locally. There are direct buses back to Jerusalem from Bethlehem, and I believe Hebron. If you can't find a direct bus back, go to Ramallah and change to Qalandia. Getting back from smaller villages use buffybotinbeiruts description. You'll probably have to ask around, but people are very friendly and will point you in the right direction. Travel between larger cities in the West Bank, especially if going via Damascus Gate or Ramallah, is a breeze.

If staying on roads controlled by the IDF, like those to the Dead Sea and Masada, use the link shuffaluff provided.

Because some of the roads in the West Bank aren't accessible to Palestinian vehicles, you may end up taking routes that seem indirect and out of the way. Your driver knows best! Trust them and ask for local advice. Looking at a map and trying to figure out which checkpoint is where and how to get from one road to another proves to be very frustrating and unduly complicated. Again, seek out local advice and trust your driver.