Located about an hour out of town, Narita is the airport that you'll fly into from international destinations; Haneda handles domestic traffic. Three types of train run in and out of Tokyo: shinkansen ('bullet train') Japan Railways (JR) trains and private trains. The bullet trains are certainly exciting, but private trains are often a better bet for daytrips. If you don't want all the stress of watching out for your stop, try a bus - prices are comparable and they often run direct. You can hop on a ferry to domestic destinations like Kushiro on Hokkaido, Kochi and Tokushima on Shikoku, and Kokura in northern Kyushu.
Once you get your head around the Tokyo train system, with its many different lines and systems, it's probably the best way to get around. Buses are frustrating: they often get gridlocked and the services end early. Taxis are exorbitantly expensive, and walking, although a fine way to explore within neighbourhoods, is difficult between them. Cycling would clearly be madness. Do as the locals do and stick to the train.