Tokyo is a wide sprawling city, with densely interconnected train networks run by a variety of operators. The easiest and most cost-efficient way to get around is by taking the subway, run by Tokyo Metro & Toei Subway, which covers almost every major spot in Tokyo. While the JR Yamanote Line covers the major spots on a loop, the subway covers those spots and all the little places in-between for a more comprehensive coverage.
In terms of the city, you can group areas into Eastern Tokyo (Ueno (上野), Asakusa (浅草), Akihabara (秋葉原, Akihabara), Tsukiji Market (築地市場, Tsukiji-shijō)), Western Tokyo (Shibuya (渋谷), Shinjuku (新宿), Harajuku (原宿), Ikebukuro), Central & Northern Tokyo (Tokyo Station (東京駅区, Tōkyō Station Ku), Imperial Palace (皇居, Kōkyo), Yanaka (谷中), Jimbocho (神保町, Jinbocho)), and Southern Tokyo (Ginza (銀座), Roppongi (六本木, Roppongi), Odaiba (お台場)). You still typically need to take a train one or two stops even within an area, as each of these neighborhoods are usually around 30 minutes apart by foot, at the very least.