Jerusalem is the largest city in and capital of Israel and of most of the monotheist religions since its establishment in 1948. Jerusalem's status is under dispute between Israel and the Palestinians. The Palestinians have claims to the city including a claim to be their capital city.
The main area where all the tourists are going is the old town, surrounded by big medieval wall. It's a very nice place to see, to get lost in the small street, see crazy historical places, but one can get very fast annoyed by the tourists and the religious fanatics. A good idea to be in peace could be to climb one of the roof to see the city from high, try random stairwells until it works.
One of the main attraction is the mount temple, it's the second holiest places for Muslims (after Mecca). It's a big temple on a big square with trees, some years ago the kingdom of Jordan offered gold to cover the temple, it's very beautiful. The problem is that it's forbidden for none Muslims during the prayers hours, and only the entrance from the Western wall is allowed for non tourists.
If you get tired of the old city there are many different things to do, like going to the Arabic side and eat a cheap falafel. You can also go to the Yehuda Market, during day time it's nice a colourful market, and at night time many bars open and young people come to play music. You can also have a walk in the Nachlaot neighbourhood just next to it, where it's nice to get lost in small streets.
Jerusalem is young and lively city, try to checkout events, concerts, festivals ...
Hitchhiking inside the city is possible to some extent. Try it it's really cool when it works. The orthodox Jews are doing it all the time.
There are different systems. There is one line of tramway that goes through the cities, it links for example the old city and the Yehuda market. It is also quite convenient to go hitchhiking outside of the city. It is possible to blackride, there are two techniques. Technique one is to buy a ticket and since the machine to punch the tickets is inside, you can punch it only if you see a controller coming and nothing can happen. The second technique is the old school one, just check out at every stops that there is no controllers (easy to spot), if there is one fly out of the tramway.
Blackriding is not possible in buses, because you have to pass by the driver as you enter. On the tram it is possible, but keep in mind there are quite a lot of checkups (maybe as much as once every 5 rides, or so). Inspectors wear white buttoned shirts in summer, sometimes with a yellow reflector vest on top, and dark blue coats in winter. They come in twos or threes, and are usually quite strict. As of September 2013, there are no "undercover"/plainclothes inspectors.
Single tram tickets are only valid for the day you bought them. There is a 90-minute transfer time for public transport in Jerusalem- that's the tram and the green buses. If you bought and used a ticket, and will not use it again within the 90-minute time, why not give it away to somebody in the tram/bus stop? Random kindness to strangers is always good.
If you plan on using public transport often, better use the electronic card. Charging rides on an electronic card gets you 20% off for single rides (for a ride you pay 80% of the price of a single ticket-ride), alternatively using the electronic card you can also purchase a daily-pass ׂ(13.5 NIS, just over the price of 2 single tickets) or weekly-pass (64 NIS, just over the price of 10 single tickets). Electronic cards. which are called "Rav Kav", are sold on most buses for 5 NIS.
There's no public transport in Jerusalem from about 2 hours before dark on Friday evening to 1 hour after dark on Saturday night or during Jewish holidays. However, buses that serve Arab neighborhoods in east Jerusalem keep running regularly.
[Hospitality exchange] should work.
You can wild camp in a corner of the big Gan Saker park, not far from the university. There are also some parks closer to the city center but there are some weird people sometimes, it might be better to avoid them.
Hostels and cheap accommodation
As a rule of thumb, the Western part of the city is much (much!) more expensive than the Eastern part, especially when it comes to hostels. In the old city and around Damascus gate you can find a dorm bed for as little as 20 NIS (4 Euro) for a night. Check Wikivoyage for more details.
A good place to camp close to the city center is Sacher Park. It's big enough to camp without being noticed, and even if you are noticed there's a good chance nobody cares. Close by, and a bit more secluded, is the Valley of the Cross, where you can also make a fire - just look for a place where people have made fires before. A bonus in this spot is you would be camping close to a beautiful 11th century monastery.
Falafel can be had for as little as 10 shekels. In East Jerusalem even for 4 shekels, but it's smaller. At Damascus gate you can get decent size for 6 shekels.
The best is to go at the end of Yehuda Market you can get a lot of leftover stuff there, veggies and fruits. And the best day to go is of course of Friday evening before Sabbath.