I come back from Jerusalem where I spent a week with friends.
10 days are more than enough to have a good overview of the old and the new city, the Mount of Olives, and even excursions outside. There is Bethlehem right next door, taxis make the round trip without much problem in less than 30 min. bypassing the wall of separation, the Dead Sea (In Gedi) and the fortress of Massadah south of the Dead Sea, easily accessible by taking the bus to the bus station (about 1h30 to 2h). Tel Aviv is less than 45 minutes away.
For EU residents, no need for a visa. Passport valid for 6 months after return.
If afterwards you want to go to other countries by necessarily friends (Syria, Lebanon, Algeria, etc.) ask to have no stamp on your passport.
Expect a thorough interrogation and a thorough search of luggage on arrival and departure from Tel Aviv Airport. Ditto in town, in most public places the metal detectors and bag searches are systematic at the entrance, no carrying of a weapon is a simple penknife is allowed in Israel. A little psychotic but we get used to it.
We rented an apartment, but there are hostels. By cons I'm not sure we can stay homestay easily unless you know someone on the spot.
It's really worth the trip even if the tension between Jewish and Arab communities is really palpable.