I would say, obviously enjoy Tel Aviv which is a cool city and go and see the normal sites, go to the great clubs and bars and restaurants especially in Jaffa but if you do, remember that only a few miles away are a load of people who can't access it all and make sure that you make an effort to go and see that.
So, when in Jerusalem make sure you visit the Arab areas. You will have to make a real effort to do it because the way the city is designed is to make them invisible. So go to the City of David 'Museum' which is a settlement and all the donations to the museum fund the settlement, then walk down the shitty road past the museum away from the old city and you will find the Arab area of Silwan. Not much to do there but if u turn left you will see the Israeli guards with their guns guarding the settlement, its a bit mad be careful and respectful (you do not want to be mistaken for a settler!) but definitely worth seeing the starkness of the apartheid, hidden in plain sight as long as you look...
Take a trip to Ramallah. Go to the Palestinian bus station outside the old city in Jerusalem and ask for the bus to Qalandiya. Get the bus, go through the checkpoint and then get a shared taxi to Ramallah via Kufr Aqab (read up on this area its v politically significant). Hang in Ramallah and go and get drunk at Snowbar in the evening. Cave bar is also good. Next day get a shared taxi to Bethlehem through Area C and you will see the settlements of kfar Adumim and Maale Adumim en route and realize the enormity and intractable nature of the conflict. See the separation wall and the church of the nativity. Then head on back to Jerusalem through the checkpoint (try and go through the checkpoint on foot) and wonder how it's so easy to travel like a normal person in Israel but so hard in the West Bank.
Also Breaking the Silence tours of Hebron are amazing if you have the time.
Then realise what the awesome, forward thinking, progressive country of Israel is really built on. And then it will be difficult to enjoy the liberty of Tel Aviv so much