From to the Airport, Tokyo
At Narita Airport there is a train station underground. For about $10US, 1 1/2 hour ride and one more train transfer - i got to my hotel in central tokyo in the Ueno district. Trying to buy the train ticket from the ticket machine is at first a little confusing but there are always helpful locals to guide you along. If you decide to take a taxi it will cost about $100US or more to get to central Tokyo.
Listen carefully: do not take a taxi from Narita Airport. Narita is quite distant from Tokyo itself; you will only waste a ton of money. There are 3 reasonable ways to get to Tokyo from Narita. 1) Take the airport limo bus. It's about Y3000 and takes about 2 hours or more, depending on traffic. 2) Take the Keiou train line to Ueno Station. I think it's about Y1600 total. 3) Easily the BEST WAY is to take the Narita Express, which is a bullet train that stops at Ueno and Shinjuku. It costs about Y3500 and takes about 1 hour. You don't have to change trains, it's spacious, comfortable and there are no problems with luggage or crowding.
Take a train/subway. They are easy to use, cheap, ubiquitous and safe. Price depends on distance. Getting from Shinjuku to Ueno (the other side of Tokyo) costs about Y190 (about US$2). The most well-known line is the JR Yamanote-sen, which circles Tokyo. It's slow, but will be useful for tourists. BTW, JR is the company running mostly above-ground trains and non-local lines. Subway is oft-times better for cross-town travel. Try to get a subway map to plan your trips beforehand (they fit in your wallet).
One problem is that cramming yourself onto one at rush hour is a pain, but an experience you will savor forever after. Another problem is that you may have difficulty determining how to get somewhere, because Tokyo train maps are somewhat complicated. Lucky for you, most of them are also written in English. Remember that trains stop running at about midnight. If you miss the last train, you're stuck for the night until 5-6am. Take a taxi only when your destination is fairly close; crossing the city with a taxi can cost you Y6000 to Y10000, especially if its late (because the rates rise). It's a truly nerve-wracking experience to watch the meter rise as your taxi is stuck in traffic...!
GETTING TO AND FROM THE AIRPORT - If you think the Japanese are smart and efficient when it comes to building airports, please think again. Narita Airport is about 80km from Tokyo, so it is prohibitively expensive to catch a taxi into town - maybe over US$400. You can get buses into the city for a mere 3000 yen (about $30) - most of them go into a terminal called TCAT at very regular intervals. Most hotels will also be served by these buses, but make sure you allow a couple of hours as the traffic can be bad.
Taxi is very expensive from the airport. We took the Narita Express from airport to Tokyo station. From there the taxi fare is may be reasonable. The taxi drivers may speak only Japanese so get a Japanese friend to write down your destination and every one will go out of their way (including the taxi driver to get you there).
The subways and the other trains are SOOOO CONVENIENT! Study that map on your first day and you will maximize your stay. Take note of the Yamanote line so you know how to get back in case you took the wrong train. The bus is also available with cheaper fares but it takes a longer travel time. The scenery though must be better.
I took the train from NRT to Ueno to check out the park and area. It took about 90 minutes (it was the slower train and less expensive). About $10 USD.
Tokyo, the metropolitan is one of the main gate to Japan. There are plenty of international flight connects to other Asia, Europe, Asian Pacific and America Continental big cities.