Just an FYI, Israel does not stamp passports anymore, they give you a card that you need to return when you leave the country.
The way countries like Lebanon know you were in Israel is if you have a visa stamp at border crossings between either Jordan and Israel or Egypt and Israel.
If you want to go to Lebanon, you cant bring Tfillin or anything with hebrew writing and you need another passport (The US and EU countries will give you a second passport for this purpose)