I was a volunteer on a kibbutz in Israel in the early/mid ‘80s. Cycling locally then was no problem but touring today might be more ‘challenging’. There is a lot more traffic and you’ll probably find a lot more no-go areas. But if you can find a route from Haifa to Eilat, via Tel Aviv, Jerusalem, Masada, Dead Sea, Hebron etc. for example, you’ll have had a great journey.
1. The one most fitting your description is the HLC route - https://www.holylandmtbchallenge.com/ I haven't tried it myself (although I plan to), but according to the description it is a meticulously planned route, making many detours to visit interesting sites, and making many creative connection between single tracks, jeep roads etc.
2. There is the Israel Trail for cyclists, similar to how there is an Israel Trail for hikers. It's still being built, with the southern sections - marked and signed - ready for visitors already. My understanding is that as the project progresses, sections of HLC will be merged into the Israel Trail.
3. There are shorter routes too, such as the "Sea to Jerusalem" route. This one I'm actually familiar with, having ridden most of it. It's a proper MTB route, with some technical sections - it visits in its run some of the most popular parts of the Israeli MTB scene.
But as mentioned above, bike touring is in its infancy in Israel, and road biking perhaps not as immediately fatal as that, but quite dangerous, whereas cycling paths are haphazard in the cities and all but non-existent between them. Most of the cycling is either groups of road cyclists, or MTB on single tracks in places like Park Canada, Beeri etc. - one day trips.