Update March 1st, 2020:
The name of the International Terminal has changed to "Terminal 3" effective March 2020.
In addition to Terminal 3, some ANA international flights will begin to use Terminal 2 starting March 29.
The station name for both the Tokyo Monorail and Keikyu trains will be updated on March 14.
Compared with Narita, Haneda is much closer to Tokyo central. There are train stations and bus stops at each terminal including international. The entire terminal is compact so it's much easier to navigate your way around.
Clearing immigration and customs is usually a breeze as there are many officers on duty to help keep the lines short.
There is only one exit, so if you have pre-arranged transportation, there should be a staff waiting immediately outside.
Otherwise, you would have the choice of either taking the trains, buses or taxis.
You will see the signs for the Tokyo Monorail on your left (including the small counter colored green for JR passes) while the Keikyu station will be to your right. For a bus or taxis proceed straight and either take the escalator or the lift down to the ground level.
The monorail is affiliated with JR. so you can use your JR pass. There is a validation counter next to the entrance though hours are limited(7:45-18:30) The monorail goes to the Hamamatsucho Station, which is a stop on the circular Yamanote Line (25 min./490 yen)
If you are staying in Shinagawa, it might be more convenient to use the Keikyu Line. Shinagawa is also a stop for the high speed Shinkansen trains, so if you have plans to immediately board the bullet trains, it makes more sense to use the Keikyu Line. A train to Shinagawa will cost 410 yen and about 20 minutes. Some trains continue service to the Toei Asakusa Line, so if you're staying in Shimbashi, Nihombashi or Asakusa, these trains will allow you direct access.
There are also services heading the opposite way to Yokohama - always check the destination to make sure you're headed in the right direction!! The wide range of destinations might make it confusing for first-time visitors. There are signs and announcements in English, but it might save you time to simply ask the station staff (Most speak English to some extent.)
Keikyu has several useful discount tickets, some of them only available to international visitors. You can find the information below
Note that both the monorail and Keikyu lines stop operating around midnight and do not run 24 hours.
Airport Limousine and Keikyu operate direct buses to various stations/hotels around the greater Tokyo area. You might find this option to be more comfortable if you have large pieces of luggage and you can find direct services to your destination.
A ride to Tokyo Station will cost 930 yen and 25-45 minutes, while Shinjuku is 1,230 yen and varying from 35-70 minutes depending on traffic.
There is service to limited locations during the late hours to cater for the guests arriving on late-night flights, however most stop operating by midnight. if your flight is scheduled to arrive after 11pm, check beforehand if your destination is served by these buses or not.
While expensive, a taxi might be the only choice remaining if your plane has arrived outside the operating hours of public transport. A ride to Shinjuku or Tokyo station might cost around 7,500 yen plus the expressway charge of 930 yen. Closer destinations like Shinagawa is likely to cost about 4,000 yen. Note all operators add a 20% surcharge for any transfers between 10pm and 5am.
Not all taxis accept credit cards, so make sure to check before boarding if you do not have Japanese cash.