Walk the walk
I personally think that one of the best ways to experience Tokyo is on foot. It's huge, yes, but you have to think of Tokyo fragmentally, as the different wards and towns that make up the metropolis all have their own vibe that becomes unmistakable after you have spent some time here; all of them can be explored individually to give a good overall picture of the layout of Tokyo.
A good route that vertically bisects the "downtown" and suburban areas of Tokyo is called "Inokashira-doori (Inokashira Ave)". This street begins at the well-known Yoyogi park, and takes a slightly winding route all the way up to the also very popular Inokashira park in Kichijoji. Yoyogi park essentially begins at the foot of the dense urban area near Shibuya, and ends in the suburban oasis of Kichijoji, which was actually ranked recently as the most desirable suburban place to live in Tokyo in terms of scenery, distance from downtown, and price.