Tel Aviv Ben Gurion International Airport welcomes you in Israel. If you are planning to travel to West Bank-Palestine, you need to pass the checkpoints controlled by Israel. While you are entering Palestine, you are not stopped or asked for your passport. But if you are passing from West Bank to Jerusalem and Tel Aviv, you will be stopped at these checkpoints. Passing the checkpoint on foot is possible.
In Tel Aviv, Allenby street is one of the most popular place. Tel Aviv port area is also nice that there are nice shops & restaurants that you can spend your time.
Middle East countries like Lebanon, Saudi Arabia and some others reject the passports which have Israel visas or entrance stamps. Carefully check before applying the visa.
Shabbat is celebrated every week from sundown on Friday to nightfall of Saturday in Israel. A kind of a prayer day. Most of Jerusalem will be closed, you can travel to Tel Aviv, everything will be still up & running there.
When you are flying back to your country, keep in mind that security checks will take too much time in Tel Aviv Ben Gurion International Airport. Better to be there 3-3.5 hours earliar. You can add more hours on it, if you are planning to pass checkpoints