Jerusalem is one of the most well known cities in the world. Tourists and pilgrims have been visiting for at least three thousand years.
Jerusalem residents ("Yerushalmim") split up along mostly religious and ethnic lines: Arab, Haredi (ultra religious Jewish), and Modern neighborhoods. Arabs generally live in the east part of the city and the western side is Jewish/modern Haredi neighborhoods weave through both halves of the city. The Haredi neighborhood of Mea Shearim, which is just north of the center of town, is very religious; it's not recommended for women to go there wearing "immodest" clothing - they've been known to get beaten up.
Jerusalem is the seat of government of Israel and representatives of most political groups are present from labor zionist to anti-zionist ultra religious to arab nationalist. Despite the fact that it's the capitol, only a couple countries have their embassies there (nearly all are in Tel Aviv). There are consulates, but they're usually not run as well. (Americans: the consulate service in Jerusalem sucks. It can take an entire day to get in and out for something as simple as renewing a passport. If you need to use it, consider going to the embassy in Tel Aviv.)
The police are generally unobtrusive, despite the security situation. Private security guards are very common - you'll see them a lot, especially on buses. They're generally well-trained and professional. Take into account that bag searches are seen as matter-of-course here: it's usual to get searched and wanded when doing something as common as going into a restaurant or a mall. Protests happen often; check the news and consider avoiding them.
There is a massive disparity in income from the poorest to the very rich, because of massive private charity networks people may have no worldly possessions to speak of they are all fed and clothed with a roof to sleep under.
Most of the population (especially in Jerusalem) speaks some English, have patience with those who are out of practice. If you are speaking English most people assume you are a Canadian, English, American, or South African and will assume you are rich.
A word to the wise, if you feel a need to spread your religion especially Christians don't do it in Israel, almost everyone has an faith even if they don't appear to practice it and they are deeply offended that you feel the need to interfere.
Jews and Arabs being descendants of Abraham have a well known tradition of hospitality. You can expect to be invited to homes after only knowing people a few hours. Try to have contacts before you arrive then you will likely never spend even a single night in a hotel. In summer it is easy to camp outside although the tourism police will catch you easily if you overstay your tourist visa.
Dumpster diving is mostly useless for foods almost everything is donated to food relief efforts, even weddings have collectors pick up left overs to distribute. If you are in the old city near the kotel you will usually be invited to dinner and lunch by a religious Jewish family.
Try the open-air market on Jaffa Street on Shabbat. On Friday afternoons before sunset, go hang out in the market and look around. It's very crowded, but empties out quickly before the sun sets. It's chaos in there and food is splayed out everywhere. Expect there to be lots of leftover. I accidentally walked away with tons of baguettes, an entire watermelon and some cucumbers that were just lying around.
Israel has socialized medicine but tourists are not covered. Cash or credit are usually taken in larger clinics, ask if there is a free clinic nearby.
Armed Forces/Deployment Avoidance Counseling
If your parents are Israeli you will be expected to do army time (2 years for girls, 3 for boys). Girls can get out of it by saying they're religious, but it's much harder for boys. People who don't do army service are expected to do a couple years of national service volunteering at places like schools or federal institutions.
Be warned: many places of employment only offer jobs to people who have done army service - even if the job is completely unrelated.
Hiking and tourism is very popular with the Israelis much more than Americans, just ask your new friends what sights are nearby. Good to see in Jerusalem are the Kotel, Temple Mount, Museum of Jewish History, and Yad Vashem Holocaust museum. Mountain biking and hiking trails fill the forest surrounding Jerusalem. Everywhere in the city there is something historical going back about 4000 years.
Most museums charge entrance fees, including the Israel Museum (which is not really worth the fee these days as it's undergoing major renovation and it's nearly all closed down). Ancient religious sites - such as the Kotel (western wall) or the Church of the Holy Sepulcher - are usually free, and worth a visit. Bring a flashlight to the Church of the Holy Sepulcher - there's barely any lighting.
There are many parks and public gardens in most areas.
Jerusalem is cooler then most of Israel because it is about 700M elevation. Summers are hot and dry winters cold with rain and occasional snow. You might think of Jerusalem as a desert city, but you'd be wrong. You don't want to be caught outdoors in the winter without warm clothes and shelter.
Traffic and parking are a mess, drivers are super aggressive, traffic accidents are the leading cause of death because everyone is late for everything. Cyclists must use sidewalks for uphill and can take to the toad lanes if they can keep up with traffic. The city is not bike friendly, though it is possible to get around on a bike.
Israeli papers are available in English if you look around. Many of the Arab TV stations transmit subtitled American reruns. Movies are usually English with Hebrew subtitles.
Many restaurants have free wireless internet, especially in areas like the city center or Emek Refaim. Hebrew University campuses on Givat Ram (near the central bus station) or on Har HaTsofim provide free wireless to anyone on campus (though only for web browsers for anyone without a uni login).
Public transport throughout the country (for major bus companies, at least) shuts down for the sabbath - from late Friday afternoon to Saturday night. Many taxi companies still run during that period, though.
The best way to get around Jerusalem is by bus, for those without cars. In Jerusalem, the bus company Egged has a monopoly over city lines, so a bus pass will get you on nearly every line. It's possible to buy bus passes good for a certain number of rides (a "kartiseeya"), or a monthly bus pass that you show the driver to get you an unlimited amount of rides (a "hofshi hodshi"). Security guards regularly patrol the bus lines and check the buses for bombs. They use behavioral profiling (such as coming up to people and greeting them and watching their reaction), so don't act nervously - just reply normally and be friendly.
Intercity buses can be taken from the central bus station (the "tahana merkazit"). Nearly everywhere in the city has a line that goes directly there. The buses are inside a mall, a couple floors up, in a big building with a clock on the front. Buy tickets from the ticket booth - you can buy either a one- or a two-way ticket (much better deal on the latter). If you're a minor ("noar"), you can also get tickets at half the price.
The intercity train system is bad, and it's usually quicker to take a bus, especially to or from Jerusalem. The municipality is working on a light rail train for the city itself, but its ever-delayed schedule a local joke - don't expect it to be finished before 2015.
Jewish young adults can sometimes get a free trip for yeshiva or seminary study. These yeshivas offer a place to stay in exchange of some Jewish study usually at a low stress level and often for free to those that can't afford it. If you blow off too many classes you might be asked to leave. If you get the boot there are dozens more you can try to get a place at.