To get a true taste of traditional Tokyo, visit Asakusa. The best way to get there is by river cruise boat. The boats run from Hinode Pier, a two minute walk from Hinode Station of the Yurikamome Monorail. The Yurikamome Line starts in Shimbashi (on the Yamanote Line). You can also walk to the pier from Hamamatsucho Station (10 minutes).
Upon arrival, look for the 'Hinode Passenger Terminal'. It is a small, new building, and the name is in English. Inside are ticket machines with English instructions. A one-way ticket to Asakusa is 760 yen.
Check this website for departure times and more information:
The cruise boat travels at a leisurely pace up the Sumida River, passing under 12 bridges. If you sit inside near a speaker, there are brief English descriptions of the bridges. If you travel during the spring, make sure to ride on the left side of the boat as the cherry blossoms are quite beautiful. The ride upriver takes about 35 minutes. Views from the boat are not incredible, but it is obviously more scenic than the subway.
If you travel in the opposite direction, another option is to get off at Hama Rikyu Garden, once the suburban residence of the Tokugawa family. After enjoying the garden, it is a 10 minute walk to Shimbashi Station. It is possible to go from the garden to Asakusa, but the boat will head to Hinode first, then backtrack past the garden. One problem with traveling from Asakusa are the crowds. There is almost always a long line of people waiting for the cruise boat. From Hinode, I've never had a problem getting a good seat by the window.
There are other options from Hinode Pier. Boats run to different parts of Odaiba, and also to the Ooi Seaside Park/Shinagawa Aquarium. Since Odaiba is in the middle of Tokyo Bay, views from these cruises are very scenic.