Always carry your passport with you until you get your Alien Registration Card (Gaikokujin Tourokushou) because it's entirely legal and encouraged for the policemen to ask to see the passport or alien card of anyone they believe to be a foreigner. In fact, I got stopped today in the train station and was asked to show my passport and Visa. You run the risk of being detained and even deported if the cops are in a bad mood, so keep it on you at ALL times. That being said, the Alien Registration Card is much more convenient and safer than carrying your passport (because replacing as lost passport quickly is hell, but the alien card is much easier). Only the passport or Alien Registration Card are acceptable forms of ID for foreigners--your school ID and other things won't be accepted.
To get the Alien Registration Card you'll need to go to the ward office of the ward you live in. You must go to the main large branch, not the smaller local branches in the ward. You'll need to bring your passport and two photos (4.5 cm x 3.5 cm). You'll also need to give your address of where you're staying in Japan, and fill out some paperwork (which conveniently has English translations) to submit it. Now, at that point tell them person behind the counter that you also need to apply for the National Health Insurance as well. In Japan, if you're here for more than 90 days (e.g. on an actual Visa and not the tourist Visa waiver) then you are REQUIRED to apply for this Alien Registration Card and the National Health Insurance. It doesn't matter if you have other insurance already--Japan doesn't care. Most doctors won't accept health insurance they don't recognize already, and some only take the National Health Insurance plan. So, apply for it because it's actually a good deal. For students with no income it's about $15 a month and covers 70% of all medical expenses while here, including dental. You can then take the remaining expenses and claim them from your own insurance when you get back home so in the end you may end up paying nothing.
You'll want to start this process early because it's a hassle and tempting fate to have your passport on you at all times. Also, make sure to go first thing in the morning right as the ward office opens because it gets really busy really fast. I got there before they opened and it still took a couple hours to get everything done and head home. They will give you proof of having applied for the alien card, but until you pick it up you still need to carry your passport at all times. It takes about 3 weeks to get it all processed and made after application. You should leave with your National Health Insurance Card the same day you apply for it and then you can immediately begin using it :)
So... who wants to see some dumb videos of me on a bus on the way to the ward office at freaking 7:30 am? Nobody? Well, I'm putting them up anyway! The first two links below are short bits of the bus ride to the huge Edogawa Ward Office in the morning, and kind of give you a feel for the other parts of Edogawa. The bottom link is a short excerpt from my 25 min walk to/from the train station everyday, while chatting with my dorm mate. We happen to live on one of the big highways in the Ward, so it's typically pretty busy. This was a fairly tame night, though. Not nearly as many people and bikes crowding the sidewalk.
Bus to Edogawa Ward Office
In other news, I went out and explored Shinjuku with a couple of Korean friends, and they took me to this awesome little noodle shop. The stuff is crazy-cheap and it's surrounded by the typical crazy Tokyo night life and bright lights.