Tokyo has two large airports: Narita for international flights, and Haneda for (mostly) domestic flights.
Tokyo's main international gateway is Narita Airport (IATA: NRT), located in the town of Narita nearly 70 kilometers northeast of Tokyo and covered in a separate article. A brief summary of options for getting there and away:
Haneda Airport (IATA: HND), officially known as Tokyo International Airport, in Ota is the busiest airport in all Asia despite being (almost) entirely limited to domestic traffic. Terminal 1 houses the JAL group including Skymark and Skynet, while Terminal 2 is home to ANA and affiliate Air Do. Shuttle services to Hong Kong, Seoul-Gimpo and Shanghai-Hongqiao use the small separate International Terminal, which is connected to the main domestic terminals by a free shuttle bus that runs every 5 minutes.
The easiest and most scenic way from Haneda to the city is the Tokyo Monorail with a station in each terminal running to Hamamatsucho (16 min, ¥470), from where you can connect to almost anywhere in Tokyo on the JR Yamanote line. The other, slightly cheaper alternative is the private Keikyu line, which has a single station between the terminals and runs to Shinagawa (19 min, ¥400) and Yokohama (27 min, ¥470). Some Keikyu trains from Haneda continue on to the Tokyo Metro Asakusa Line, providing one-seat rides to Nihonbashi (38 min, ¥590) and Asakusa (45 min, ¥640). Trains also continue all the way to Narita Airport, but for airport to airport transfers the Limousine Bus will be faster. Note that there is no JR service to or from Haneda, and the "Airport Terminal 2" station that pops up in some route search engines refers to terminal 2 at Narita Airport, not Haneda!
Expect to pay anywhere from ¥4,000 to ¥10,000 for a taxi to central Tokyo.
Chofu Airfield (Chofu hikojo) serves only some turboprop flights to the Izu Islands south of Tokyo. The nearest railway station is Nishi-Chofu on the Keio Line, a 15-minute walk away. Alternatively, you can take a bus from Chofu or Mitaka stations.
There is a frequent intercity Shinkansen service to and from Tokyo Station (Tokyo-eki) in Chiyoda, from where you can easily connect pretty much anywhere in the city on the JR Yamanote or Metro Marunouchi lines. For all northbound trains, you can also hop on at Ueno, while all westbound trains stop at Shinagawa.
For non-Shinkansen services, Shibuya and Shinjuku stations offer local connections to the west. Ueno and Ikebukuro stations connect you to the northern suburbs and neighboring prefectures.
While you can drive into the city, it's really not recommended as the city can be congested, signs may be confusing and parking fees are astronomical.
Hitchhiking into Tokyo is pretty easy, but hitchhiking out is considerably more difficult. It's definitely possible for determined cheapskates though, see Hitchhiking in Japan for a detailed list of tested escape routes from the city.
Highway bus services link Tokyo to other cities, resort areas and the surrounding prefectures. There are JR and private bus companies. Bus service may be cheaper, but the train is probably more convenient. If you have a JR pass, then you should generally stick with the trains.
Long-distance buses use a number of terminals scattered throughout the city, but the main JR depot is at Tokyo Station's Yaesu-minamiguchi exit, while Keio and some other private companies use the Shinjuku Highway Bus Terminal, opposite Yodobashi Camera near the West Exit.
One of the great ports of the world, Tokyo also has domestic ferry services to other points in Japan. However, none of the regular international ferries to Japan call at Tokyo.
The main long-distance ferry terminal is Ariake Ferry Terminal, located on an artificial island adjacent to Odaiba in Tokyo Bay. The nearest station is Kokusai-Tenjijo-Seimon on the Yurikamome line, but it's still a bit of a hike. You can also take a direct bus from Shin-Kiba station on the Metro Yurakucho line. The main services from this terminal are:
Ferries to the Izu and Ogasawara Islands leave from Takeshiba Terminal, adjacent to Takeshiba station on the Yurikamome line. Cruise liners tend to use the Harumi Terminal, best accessible on bus ?05(To-05) from Tokyo station Marunouchi South Exit or (Tou-12) from Tokyo station Yaesu exit. International ferries and cargo ferries that also take passengers can leave from other terminals too, enquire with your shipping company.