Hello!
In Israel the week starts on Sunday and religious events with sundown. The day of rest is Friday evening to Saturday evening, the new year will begin with sundown, etc. That means most shops close early afternoon on Friday - most will open up again on Sunday, some Saturday night. Same for public transportation - stops some hours before sundown on Friday, starts again after sundown on Saturday. Same with holy days.
In the bigger cities, you will find more establishments that do not adhere to the religious schedule. Between these cities (Jerusalem - Tel Aviv, Tel Aviv - Haifa) you will find sherut services - a van like a guagua or porpuesto in Latin America if you are familiar with those - that operate on Saturdays and holidays. The one exception is Haifa, where stores, museums, and transport don't observe the sabbath. Arab towns and arab establishments in cities are also open.
Shuffaluff - do you mean Nativ Express as the 'Arab' bus company? If you do, they do not run on Saturday nor on holidays - they are part of the public bus system just like Egged. So do not count on this for transport.
FOR THE NEW YEAR/SATURDAYS:
In Tel Aviv you can move easily by bicycle. It is more secular, so you will find a lot of restaurants open and of course the beachfront. And Yaffo is open as well (the Arab part of town). In Jerusalem, one part of the town is totally dead (the super religious) and then in the old city you can find things open because of the Arab population. So it is an interesting contrast. You'll have to walk to see that contrast (and not as flat as Tel Aviv)
Probably Eilat is like Tel Aviv - stuff is open. THat is a guess though - so check that better if you are interested in that option.
This is the best service to check for how to get around - they have an english page and compiles information from the different bus companies and the trains.
http://www.bus.co.il/otobusim/Front2007/homepage.asp?LanguageID=20&Design=2007
I'm not sure about a sherut out of Nazareth...
Hope this helps!