It is a good idea to hire a private guide for a day to tour the Old City. "Sandemans" has a free introductory tour that you should take on your first day to orient yourself. You can go to other sites on your own - public transport is good.
An organised tour is the best way to go to The Dead Sea. You cannot go to all the sites in one day via public transport.
The Abraham Hostel has a day tour to the Dead Sea. This is transport only, no guiding. See here:
http://abrahamtours.com/tours/masada-sunrise-tour/
100, you write that you want to visit Bethlehem and Jericho, right? I'd suggest going on a Bethlehem day tour that covers both. Egged Tours has one, but you can check with other companies as well. Basically the companies offer a thousand options for tours in Jerusalem, such as walking tours, tours of the City of David etc.
It's well worth it, IMHO, because you get so much more out of it then by just traveling around on your own.
Good luck!