First Tokyo visit - how many days?

I would assume the biggest question you have is "How many days is necessary to see a big city like Tokyo?"

Of course you can't see everything, but to get a sampling of the traditional sites plus the neon/craziness/chaos of the city,

Three full days in Tokyo seems sufficient if only visiting the major sites.

For those not used to seeing so many people in one place (We have 13 million people living, just in the central city...), you might want to use your third day to go to quieter places outside Tokyo. The ancient capital of Kamakura, the World Heritage sites of Nikko or the beautiful scenery at Hakone (and hopefully a view of Mount Fuji!) can all be visited as a daytrip from Tokyo.

For those completely lost, I have arranged a rough itinerary so you can see how much you can cover in three days. 

72 Hours in Tokyo

And just brief advice on your overall itinerary...

From my experience in the travel industry, I have observed that first-time visitors to Japan spend roughly 7-10 days here, with the second half of the trip centering around Kyoto and the surrounding cities of Nara and Osaka. 

With the opening of the Hokuriku Shinkansen high-speed rail that heads to the Sea of Japan side, Kanazawa has become a popular destination. The Hokuriku route serves as an alternative to the "Golden Route" of Tokyo-Hakone-Kyoto which is the more common routing along the Tokaido Shinkansen. 

With another few days, the visit could be extended to include Hiroshima and the Miyajima Island.

About 85％ of Japanese travelers opt for the Shinkansen if traveling between Tokyo and Osaka. However for travel between Tokyo and Hiroshima, the Shinkansen share drops to around 65%. For Fukuoka (Hakata), more than 90% choose to fly as the Shinkansen would take around five hours.