I like your suggestion. You are right, I don't want to risk taking an overcrowded bus with a tired family. I will probably look to get a car and driver rather than attempt renting and driving ourselves.
What we have planned is to stay in Tel Aviv for the first five nights. Around that time we will go to Galilee and Haifa but come to Tel Aviv to spend the night. Then we move to Jerusalem where we spend a whole day and a night. Also cover a little bit more of Jerusalem the next morning and then head towards EIn Bokek and the Dead Sea. Hopefully if the weather helps, we can hike up to Masada and then head out towards Eilat. Does that make sense? I know its hectic but there is so much to see and such little time.
If you use Tel Aviv as a base, it is fine for touring Haifa and Acre which you can get to easily by train. It is not the most convenient for seeing the Galilee as you will end up spending a lot of time "commuting" back and forth. It would be better for you to spend at least one night up north.
You are still leaving very little time for Jerusalem - in essence, barely one day total - when there is so much to see in Jerusalem.
Driving in Israel is easy. If you want to get a car and driver to get to Eilat be aware that is going to be expensive. You may be staying in Eilat, but the driver has to make his way back up north. It would also be a good idea to try and find someone now. You will be in Israel during a very busy time and not every cabbie is willing to make that trip.
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