The best experiences for me always occurred when I was relaxing and exploring in Jerusalem: the Old City shops (especially the Christain Quarter), Ben Yehuda Street, Kotel (Western Wall) and the Mount of Olives.
Quick Tips:
Depending on the time of year, it can be very hot in Israel, but if you are planning on spending any time in a holy site (church, synagogue, mosque, Western wall, etc) you should be prepared to "cover up". For men, always have a pair of pants that you can pull on quickly. For women, bring a pullover/wrap around skirt and a shirt (if you are wearing sleeveless). I always kept a wrap around skirt in my bag and then threw it on when I wanted to visit a holy site.
When making any kind of travel plans on your own, be very aware of the Israeli/Palestinian borders. For example, you cannot use an Israeli phone card in most PA Authority area. Also, PA areas and Israeli rental cars have different license plates. It is not allowed to drive an Israeli rental car in a PA area and vice versa. You can take a rental car to the border and take a taxi into the PA authority area.
If you are traveling with a tour guide, it is not something to be too concerned about. They are very in tune with the political situation and will typically know where is safe to travel and would not put you at risk.
Best Way To Get Around:
I used the bus system heavily; however, it is a decision that you have to make for yourself. I personally still use the bus station, but others do not.
You can also take sheruts (shared taxis) to other towns.
Taxis are available all over Jerusalem.