The Jews have over four thousand years of history, but Israel is only a few decades old. To state the obvious, Israeli culture is the product of an old Jewish legacy and a new government.
Israel has few natural resources, and Israelis pride themselves on being a self-made people. Although they know how to work long and hard (only the Japanese have a longer work week), they also enjoy their leisure – e.g. on a beach day or a stroll along the board walk on Shabbat (the Jewish holy day) is common. Israelis are also avid sports players, and it's not uncommon to see them playing tennis, football, basketball, swimming, or running on Shabbat.
Israelis are community-oriented and value relationships. They are very hospitable when you get to know them personally, but they sometimes appear a bit rude to outsiders. Israel has had a little more than half a century to establish itself, and Israel's citizens have to be aggressive if they want to get things done.
Many Israelis want to get the most out of their time, and they will often treat strangers with impatience. It is not uncommon for an Israeli driver to cut in front of you or a fellow passenger to elbow you out of the way while getting on a bus (to name a few examples).
Native Israelis pride themselves on being sabras, or nationals who were born in Israel. They took the name from the sabra cactus, whose pear-like fruit has a tough, prickly outer shell but a sweet inside. Many foreigners find the same thing to be true of Israelis.
Unlike in some other cultures, many Israelis are sincere when they tell you to come over for a cup of coffee. Israelis won't mince words and will tell you exactly what they think of you. They are being direct and honest, and if you don't say what you think, they will think you are being sneaky.