Q: Which airport should I use to fly in and out of Tokyo?
A: There are two major airports servicing Tokyo, Japan - Narita and Haneda.
For many years, Narita used to be known as the major International Airport and Haneda was more of a domestic airport, but in the past five years, this has changed significantly. Nowadays, it is more common to fly into Haneda than it is to come through Narita.
Haneda is now considered an International airport and is even a tourist destination in it's own right. The best advantage for flying in and out of Haneda is that it is only 30 mins from downtown Tokyo and is accessible by taking a local train or airport bus. Narita, on the other hand, is a good 90 minutes outside of Tokyo and the fastest way into the city is via the Narita Express or Keisei Airport train. The cost of the Narita Express is about 3200 yen or close to $35 CDN approx.
Not only is it faster to use Haneda airport, it' also cheaper than Narita. Here's Japan Guide's write up on the how much it costs to get to Tokyo from Haneda.
http://www.japan-guide.com/e/e2430.html
While you can take the train from Haneda, it is still a burden if you are pulling luggage onto the local trains, especially if you haven't travelled to Tokyo before. Misterpiggy recommends taking the Airport Limousine Bus. It has several drop-off/pickup points throughout Tokyo and is fast and efficient. The cost, for example, to travel from Haneda Airport to Shinjuku is 1230 yen one way, which is currently about $15 CDN. You can find more information about the Airport Limousine Bus service routes and rates here:
https://www.limousinebus.co.jp/en/platform_searches/index/3/16
The Bottom Line
If it's possible, try to fly in and out of Haneda Airport, it's closer to Tokyo, therefore being a much faster and cheaper option then Narita which is farther away and more expensive.