Train to Narita - The easiest way to get from the airport into Narita is by train. The Keisei line has frequent trains from the basement of both terminals into Narita. The cost is about 250 Yen. You can buy your ticket from the ticket booth or from the ticket machines. Going in to town the trains begin at Terminal 1, stop at Terminal 2 and then Narita town. Coming back, make sure you get a train from Narita town to the airport terminals. Some trains from Narita go only to 'Higashi Narita' (East Narita), an airport stop for airport employees, not travelers.
Bus to Narita - There is also bus service between Narita and the airport. Enter the bus through the front door and take a paper ticket from the machine. The number on the ticket tells where you boarded the bus. A number board on the front of the bus tells how much you'll have to pay if you get off at the current stop. You must use exact change. However, most bus fare machines will change a 1000 Yen note. (Remember that after you put in the note you will get 1000 Yen in change. You still have to pay for the ride from that change.) Buses can be crowded with commuters and school students and confusing for the uninitiated.
Tourist Bus in Narita City - Narita City operates a visitor-oriented tour bus that makes two routes around the city. More information here: Narita City Round BusTrains and Buses to Tokyo and Elsewhere:
For detailed information about express (somewhat expensive) trains and buses into Tokyo and environs, click here for info from J Sachs.
For more information on the train system and a different view on trains from the airport into Tokyo and environs, click here for info from Ken Heard.
Some useful rail links from Widya Santoso: Here's a link to the greater Tokyo JR network (PDF file), which includes the lines to Narita. This link allows one to calculate the schedule and fares between two stations.
Widya also reports that JR East is now offering a special Suica + Narita Express fare. For 3500 Yen you get a one way ticket from Narita to Tokyo plus a 2000 Yen Suica card which you can use for transportation and some purchases withing metropolitan Tokyo. "Basically for just Y500 more of the price of a standard N'EX ticket you get the same ticket for Y1500 and a Suica card with Y1500 credit (and Y500 deposit fee). This can only be purchased from the JR Offices (where one goes to have JR Pass vouchers redeemed) at Airport Terminal 1 and 2."
Tip: if you have a JR Rail Pass voucher to validate and are also interested in purchasing the Suica/N'EX fare and plan to travel within Tokyo using the Suica card on the date you land at Narita, when validating your pass be sure to ensure that your pass is validated for the next day, not the date you purchase/use the Suica/N'EX fare. If you purchase the Suica/N'EX fare and validate your JR Pass for the date you land you use up a day of the pass. You also get an 'extra' day out of the pass by not using it to travel from Narita to Tokyo.
Of course, if you are going to use the pass to travel elsewhere within Japan or you don't plan to purchase the Suica/N'EX fare, validate the JR Pass from the date you land.
Look here for information on getting to: Haneda (domestic) airport.