Here's a quick list of some Israeli music on YouTube that I've liked over the years, and continue to like.
Next for Arik, The Churchills...pretty much a copy of the Beatles, but toned down a bit.
Achinoam doesn't know
From Lul, a classic comedy series with Arik Einstein, Uri Zohar and others around 1972.
I like to Sleep
This is one of my favorites, also from Lul.
Why do I take to Heart, Arik Einstein and a very young Shlomo Hanoch.
A relatively recent group. Great pictures in the video, though.
Ethnix, She won't come back.
Yehuda Poliker songs. He's of Greek heritage, worth learning about his family history. Early music (Benzine) was straight on Israeli Rock. Later he let the Greek influences out.
Of course, there's always Aviv Geffen. I never really got his goth look in the early days, well, still..., but with his parentage you knew he could do something like this. Cry for You