The Tokyo subway system is without a doubt the most efficient way to get around the city. (Not only are taxis unbelievably expensive; they also get caught in the tangle of traffic.) True the system is complex. The secret to using it is to know the color code of the line that stops closest to your destination. If you have one of the free color-coded maps with you you’ve solved the majority of the problem. The stations are not always convenient (expect a lot of steps not a lot of escalators) and the ticket machines can be baffling at times but the attendants will be as gracious and helpful as they can. (We found it best to have something for them to look at a map say with either English or Japanese words. They were not as skilled in understanding spoken English.) Try to avoid the subway at rush hour however unless you want to see the legendary pushers literally pack as many people on the trains as possible.
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Tokyo, other things to see
Besides the areas in Tokyo described on your website, you musn't forget to add: Odaiba, the fairly new shooping center in the middle of the harbor, with an excellent view of the skyline of Tokyo. Take a boat from Asakusa, the famous Sumida River Cruise (not that spectacular, but hey, you do get a relaxing change of pace and a different angle on the city).