Definitely rent a car if you are going to go outside of Jerusalem. You can accomplish a lot more that way
Like stated above Ein gedi, masada (would do Sunrise and definitely in the summer!) and dead sea can be done in one day. Ein gedi actually has the the generic one hour hike that everyone does to full day 8 hour hikes (and you will hopefully get lucky and get to see some ibex!)
The bullet factory is really cool and a great off the beaten path stop (central Israel)
Rosh Hanikra is really beautiful and worth a stop. Thats up north literally on the border of Lebanon in the galil
If you are there only one shabbos has to be Jerusalem. For the second shabbos I would say toss up between Tzfat and Chevron.