Here's some general info, as well as some hints and tips I've picked up:
Trains and Subways:
Platform sign for the Yamanote Line around Tokyo.
This website tells you how to get from one station to another, all you need to know are the names. It gives you different options, prices, and the timetables of the trains. Basically, it is a life saver and I use it all the time here.
Now, there are some things you should know. In Tokyo, everything is in English (including the platform signs and all the main trains have English signs and announcements for the next stop), so don't worry about that. Also, there is no such thing as a return ticket - everything is one way. This also makes it more expensive. And you should know the trains/subways aren't cheap. On a typical day going to and from any place will cost you about 1000yen.
The trains are cheaper, especially the JR ones (which are most of the ones around Tokyo). Tokyo Metro and the private subway lines always seem to be more expensive, so if you can, try get the trains for all or most of your journey.
The trains and subways run from around 5am - 12am. If you are caught without a train, there are hundreds of 24 hour cafes and restaurants on every corner you can just go into and sit down/sleep. I believe the Japanese are usually far too polite to come and wake you up.
The trains and subways aren't as crazy as you see on the internet, with a hundred people squished against the glass. Mostly. Weekdays from 7-9am is the only time you will see this, and I've been on these trains, and it is not a pleasant experience. Going home around 6-8pm is quite busy, but not near as bad as the mornings.
And don't worry about being groped - you probably won't be. I've been here two months and it's never happened to me or anyone I know, and talking to women who have lived here for years they say the same thing - it's not as common as the media would have you think.
Don't ever expect to get a seat. It is a rare and joyous occasion when this happens.
The Suica and Pasmo cards:
If you are going to be in Tokyo for more than a few days, I would recommend one of these. From what I understand these cards are both the same, but since I have the Suica card I'll give you more detail on that. Basically it is a type of credit card that's topped up with money, and then used to get the trains/subway. It is much easier and faster to just scan it as you go onto the platform than having to calculate fares and buy tickets each time. You can also use it to buy things at a lot of vending machines, as well as some supermarkets.
The Suica card cost me 2,000 yen, but this was with 1,500 yen on it to use, so really the price is only 500 yen, which is very cheap. It also gives you an option to put your name on it so it can be returned to you if lost (or they will give you a new card with the amount you had on the last card). I bought mine at a ticket machine in central Tokyo, although I am not sure if every station has these machines. Every station has machines where you can top up the card, and if there's a problem (such as you have no money left and the barriers won't let you leave. . .don't worry it happens often!) you can go to the help desk and they will sort it out. It's free to top up and use the card, so I'd see no reason for not having one (although my friend still doesn't, unbelievable!).
Car, bus, taxi. . . .
Driving in Tokyo is madness. Don't do it.
I've never gotten a taxi but they are supposed to be pretty expensive, and only to be used when you need to (such as if the train isn't running).
I've never gotten a bus either, but I always see them around Tokyo, so it definitely could be an option.
Bikes are pretty cheap here and you always see people cycling around the outskirts of Tokyo, not so much in the middle of the city. You can't put a bike on a train subway or bus though.
Walking
As with every city, you will walk until you can walk no more! Unless you're like some of the local girls and can walk comfortably in your six inch heels, comfortable shoes and blister plasters are a must. Crocs are very popular in Japan, especially during the summer, and now I know the reason why.