The first two weeks of June is a perfect time to visit Israel. In early June the serious heat of the summer usually has not yet set in.
As you were advised above, it would be a mistake to use Tel Aviv as your base for the duration of your trip. If you plan to tour independently (or integrate days with a guide into you itinerary), I suggest the following bases and in the following order, leaving Jerusalem for last:
3 nights Tel Aviv - Consider a hotel in the Atlas Group (www.atlas.co.il)
1 night Haifa or Western Galilee
3 nights Galilee - Consider the Pastoral Kfar Blum, Hagoshrim, or Kibbutz Lavi kibbutz guesthouses, or the Ramot Resort.
4 nights Jerusalem
From Jerusalem, you can do a day trip to Masada and the Dead Sea. You'll head directly from Jerusalem to Ben Gurion Airport for your departure.