For all your public transport questions, see here. This "umbrella" site covers all the bus companies as well as the trains. The train ride to Jerusalem is very scenic, but it's extremely slow and the Jerusalem station is in the back of beyond. Take the bus, which goes every 15-20 minutes.
Where to go when depends to some extent on the day(s) of the week. There is no public transport on Shabbat - the Sabbath - between sunset on Friday and after dark on Saturday, so unless you rent a car or bicycle you won't be able to travel around on those days. One exception is the sherut (shared taxi) between Tel Aviv (the city, not the airport) and Jerusalem, which runs 24/7.
Everything in east (Arab) Jerusalem functions normally on Shabbat, and if you're interested in going to Bethlehem or Ramallah Saturday is a very good day to go. Buses to Palestinian cities leave from outside the Damascus Gate of Jerusalem's Old City.
It's actually easier to get to Jerusalem than to to Tel Aviv from the airport. The Nesher shuttle will take you from the airport to any address in Jerusalem for about NIS 60, and it goes 24/7. It's a 10-seater minibus and goes when it's full, and it fills up quickly. There is nothing similar to Tel Aviv. There are trains from the airport to Tel Aviv (except on Shabbat), but the chances of any of the railway stations in Tel Aviv being near your accommodation are remote. For returning to the airport from Jerusalem, you need to call Nesher the day before. The telephone number is (02)6257227, and it will pick you up anywhere in Jerusalem.
Skip Eilat on such a short visit. It's a long way to go - five hours in each direction - for a beach resort. Tel Aviv beach is much nearer. Public transport in the Golan and Galilee is sparse, and geared to residents rather than tourists, meaning that buses leave the villages in the morning and return from the big town in the late afternoon/early evening. Not a lot of help to you. Consider an organised tour for that part of your trip, or rent a car.