It all depends on where you want to go. It's arguable that Tel Aviv and Jerusalem have the best transportation in Israel, but more importantly, these cities, especially the more popular and touristy parts of them are a nightmare to drive in. My advice would be to not take a car for your stay in Jerusalem and Tel Aviv.
It also depends on where you want to go to in these cities, but you might even eliminate the need for any transportation, depending on where you book your hotel. This is especially true for Jerusalem, as most of the tourist attractions are located close by to one another.
You mentioned that you want to travel to the north as well, there is becomes more problematic to rely on public transportation. Again, this depends on where you want to go, you can take a bus or a train in the morning to a major place like Haifa, Acco or Nathareth and if you plan to spend an entire day or more there you can get by without a car. But if you plan on visiting many small places which might have very infrequent service and you are on a tight schedule, then you will probably have to get a car.
Regarding Holidays the Sabbath and public transportation:
Taxis work all year round except for Yom Kippur, where there is almost no traffic at all, including private cars.
There is no public transportation in the Sabbath in Jerusalem and Tel Aviv* and there is reduced public transportation in the north. The Sabbath starts on the evening of Friday, and ends on the evening of Saturday, the exact times change during the year, and so do the start/end times of public transportation. The same goes for Jewish holidays, you should check if any holidays fall on your expected visit days.
You could probably get by without a car if you plan to visit major and big cities and attractions, and it would probably be easier without a car. If you plan on visiting many small places then you would probably need a car.
* - Lately a new program was started by the municipalities of Tel Aviv and several neighboring cities of operating free public transportation on the Sabbath. This is a very new initiative and I don't know if it'll hold and if so how this service will look in the future. Right now it's not something I would count on several months in advance.