NOTE: While I am using a conversion factor of approximately $1/100 yen (it's the quickest way to convert), the exchange rate as of 4/2/12 is around $1/82 yen, meaning that prices are 20% more than what I'm quoting here.
1. How do I get around?
The nice thing about all of the train lines in the Tokyo area is they are very English friendly. The stations and ticket machines all have lots of English so that you can know what to do and where to go. In addition, trains run early (around 4 am) and late (finishing at around midnight) so you won't get stranded.
One word of caution: I wouldn't use a cab unless I had to. While they're great to get a picture of, the taxis are extremely expensive. To get in it will cost you $6. To travel 1.2 miles (2 kilometers) it costs $12. Meanwhile the comparable train ticket is $1.50.
2. What about the airport?
Narita International Airport is about 90 minutes by train from central Tokyo. If you were to take a cab, it would cost around $300, so I would recommend one of the following options. I've only done this 3 times, so my traffic times may be wrong, but I think I'm relatively close.
Option 1: JR Narita Express. If you are going to the Tokyo station, Shinjuku, or Shibuya areas, this is often the best option. If you are using some sort of JR pass, it's an even better option as it is covered by the JR East Pass and JR Kanto Area Pass. From the airport to the station, you're usually around an hour out.
Option 2: Keisei Electric Railway. This is a great alternative if you're in Ueno or Shimbashi or if you don't mind a transfer with luggage to another train. They are less expensive than JR if you don't have a pass or will just be using a Tokyo Furii Kippu.
Option 3: Friendly Airport Limousine Bus. Traffic along the route can get heavy, so the timings may be off, but it should take about an hour and 45 minutes and they have some great time listings on their website.
3. I want to see some interesting people. Where do I go?
One of the great things about Tokyo is the different people you come across. If you're looking for the widest variety of interesting folks though, there are really 2 areas that you must visit.
1. Shibuya. This is ground zero for the gyaru culture, which really has to be seen to be believed. Most of the action is on the northwest corner of the station. It's home to Hachiko, the largest intersection in the world, and something that really puts Times Square to shame. The largest outdoor video displays in the world are found here. The area as you walk towards the HMV sign is a veritable cavalcade of people and interesting things.
2. Harajuku. Harajuku is...hard to describe. Kind of like Shibuya, but kind of not, they have some amazing people here too as well as some great side streets. If you cross the train tracks, you walk right into Meiji jingu, the biggest Shinto shrine in the world. It was rebuilt after World War II, but it's still very interesting to see, and most of the time it's very peaceful.
4. I want to see some interesting things. Where do I go?
Tokyo has an interesting mix of things to see. One thing to note though, is that most of the "old" stuff in Tokyo is reproduced as a result of the city being destroyed by an earthquake in 1923 and then firebombed in 1945. As a result, cultural heritage is not something it has a lot of. Nevertheless, there are still some parts of old Japan that survive, though you may want to consider a day trip (described below) in addition/instead.
Consider a trip to Asakusa's Senso-ji temple and environs. It's nestled in the middle of standard Japanese housing and buildings, but it is a very cool looking area. In addition, you can find some of the best souvenir shopping around in the mall leading up to the temple.
Imperial Palace. The palace was untouched by World War II, but unfortunately you don't get to see much of it. It's a brief walk from Tokyo station, with its 1900s facade. They do have an amazing bonzai garden in front of the palace as well, if you like that sort of thing. If you are in this area, consider a side trip to the Tokyo International Forum, where Trey Ratcliff got this shot.
Meiji Shrine. See above!
Besides the aforementioned Shibuya and Harajuku, consider a couple of other destinations.
First is Ginza. There is some amazing (albeit expensive) shopping, but Ginza is one of the best places in the city for seeing the modern side of Japan. It isn't as garish as some other districts, but a little more toned down.
Akihabara. This is the electronics capital of the world. Tons of garish lights to distract you into their stores to spend lots of money. In addition, they've got an amazing market below the train tracks where tiny vendors sell electronics scraps like transistors, vacuum tubes, and all sorts of other good stuff.
Shinjuku. If you like skyscrapers, this is the best part of the city for them. It's the most stable ground in the area, so as a result most skyscrapers were built here. Some interesting buildings of note are the Tokyo City Building, the Park Hyatt Tokyo, and Mode Gakuen Cocoon Tower (shown here)
Roppongi. More foreigners spend their time here due to the proximity of the US Embassy, and as a result, there is a lot to see. Be cautious, as it's not a criminal area per se, but it is one where you may want to avert your eyes from time to time as it's a prostitution and seedy nightlife hotbed. They do have Roppongi Hills (a 55 story building with a great cityscape view - if you're looking for sunset shots with Mt Fuji in the background, this could be the place to get it. It's about $10 to get to the observation deck) and Tokyo Tower though, so it's worth a trip in.
Inokashira koen. If it is still hanami (cherry blossom season), this is the place to see and be seen in Tokyo. It's outside the south exit of Kichijoji station on the JR Chuo line.
5. I want to get out of the city for a day. Where should I go?
There are 3 very doable daytrips out of Tokyo and one that is more difficult that are worth considering.
1. Kamakura. Kamakura is the former capital of Japan and is unscathed despite the calamities that have befallen Japan in the last 100 years. It's approximately 90 minutes from Tokyo and it's got some amazing buildings and people seeing that you will not get in Tokyo. If you want to see Japan as the tourist brochures see it, this is the place to go. Consider traveling in via the JR Yokosuka line from Tokyo or Shinagawa stations, getting off at Kita-Kamakura, and walking past some of the best Japanese shrines - Engakuji, Meigetsuin, Kenchoji, and Hachimangu, on your way to Kamakura station, where you will take the Enoshima line to Hase, where you can see the daibutsu (a huge metal buddha statue) and Goryo shrine, which has some great views of the city.
2. Nikko. Nikko is more difficult to get to than Kamakura, but if you don't mind, it's amazing. You can take the bullet train to Uchinomiya, then transfer to a small local line to Nikko if you have a JR pass or take the Tobu Nikko line from Asakusa to get there. Either way, it's about 2.5 hours outside of Tokyo, so it's a bit more time consuming. If the cherry blossoms are gone in Tokyo though, you have an excellent chance of seeing them here. The highlight of Nikko is the UNESCO World Heritage Site that includes the shrines and temples built by Tokugawa. Out a bit further is the Nikko National Park, which has some amazing sites as well.
3. Mt Fuji. Unfortunately I've never done this, so I can't speak to it, but it's supposed to be phenomenal. You can take the bullet train from Tokyo to Odawara and then take local lines from there into the park.
Japan Rail Pass. It's $340 for 7 days, but you'd spend that on one bullet train trip.
6. I want to eat. Can I do that?
Eating is easy in Japan! You can eat any non-Mexican food at any price point you want. You may want to consider a yakiniku (grill your own meat) place - though if you do, do it at lunch as it's significantly cheaper. Yoshinoya is a chain of meat bowl places that is available around the country, and they are really good. In addition, there are vending machines and convenience stores on almost every corner, and they have a lot of things available to satiate your hunger.
See my earlier post on this topic!