Is it safe to travel to Israel?

No place in the world is 100% safe but Israel is a relatively safe country to travel in. In the past few years the security situation in Israel has been increasingly calm since the end of the Intifada. As a result, tourism to Israel has been growing steadily and last year alone approximately 2 million tourists visited the country.

Are tourists allowed to enter areas outside of the Israeli responsibility (Palestinian areas)?

Passage to the two major tourist cities of Bethlehem and Jericho in the Palestinian Authority is direct without prior clearance or required authorization.

As regarding the rest of the Palestinian areas, it is recommended to forward requests to the IDF Public Relations Office, Fax: +972-2-5305724. All requests should include: Name, passport nationality and number, destination, name of departure into Palestinian Area Crossing if known; if crossing is by car – name and details of driver as well as car registration number. Fax replies will only be sent to Israeli tel./fax numbers.
From The Israel Ministry of Tourism