Israel is generally a safe country. Crime levels are low and generally people are warm and friendly.
I have hitchhiked extensively throughout the country and only felt unsafe only once, in the South when there was a group of big aggressive dogs.
In the Palestinian areas of the West Bank it's important to not speak Hebrew or show pro-Israeli signs. It's not really a problem to be Jewish if you don't show that you're Jewish.
The West Bank is divided in three different zones. Area C is territory under Israeli control, which is practically like Israel except the higher frequency of check points, area B is under Palestinian civil control and joint Israeli-Palestinian security control. Area A (about 3% of the West Bank) is under full civil and security control by the Palestinian authority. It's illegal for Israeli citizens to enter area A. (See also the Wikipedia article about the areas)
I have hitchhiked in all 3 different zones in the West Bank and even ended up using my knowledge of Hebrew (which is far better than my Arabic) in such cases after I felt comfortable enough with the driver.
Gaza is not considered safe and only really accessible for journalists and aid workers. Check Wikivoyage for more information.
Avoid demonstrations of any nature, even more so in tense times.
Avoid discussing politics as much possible.
Be careful in the West Bank with showing Jewish and Israeli stuff and in Israel with Islamic or Arabic clothing or items.
In times of higher tension
At times rockets are shot by the Hamas in the South and Hezbollah in the North once in a while. As of 2014 only the far South (Eilat) has not been in the range of such rockets. At times the frequency of attacks can increase, which can cause a lot of stress, but even in such cases air alarms and shelters will provide plenty of safety. In general the number of civilian casualties in Israel remains extremely low and even through traffic is not that bad you're more likely to suffer from a car accident.