From BenGurion Airport:

The quickest way is to buy a seat in a shared taxi (sherut). Taxi ranks are on your right hand side as you exit customs with your baggage.

Second-best: catch  a bus at the bus stop across the street.

NO trains go as yet to Jerusalem from the airport.

From Tel Aviv and other points: regular bus services, also trains.

 

The Central Bus station is in north Jerusalem, the Railway station in the south of the city.

The Old City with its holy places lies to the east.

West Jerusalem is the more modern section.

 

Hitchhiking is possible but not recommended.

 

The train journey from Tel Aviv to Jerusalem takes about an hour and a half . 

Although slower than the bus (55 minutes) it is much more scenic and covenient if you are not in a hurry. Buses also get stuck from time to time in traffic jams but  trains do not ,so paradoxically-like  in the hare and tortoise  story, the train may be  the better choice.... 

The new highspeed train is under construction but not slated to open until 2020 (maybe).

If you are going from Jerusalem to the airport the best way, especially at night, is by Sherut.  Call to reserve your time and they will pick up at your door and take you to the airport while picking up a few others along the way.  They operate 24 hours a day and the set price is 55 shekels (under $20.00).  Nesher Taxi 1599-500-205 or 625-7227.  The operators speak English.  You need to book a Nesher Taxi at least 48 hours in advance.  The office is closed on Friday afternoons from 12noon to Sunday morning at 7:00am so if you are leaving on a Sunday or Monday, be sure to book your taxi on Thursday or Friday morning.