To travel around Tokyo, the best way is to use the subway system. Purchase a Suica or Pasmo card for convenient and cashless travel. Trains can get crowded during peak hours, so it's best to avoid traveling during those times. For day trips to Kamakura and Nikko, purchase a JR pass for discounted train fares. Taxis are also available, but can be expensive.
If you have more time, consider adding a day trip to Mt. Fuji or the charming town of Hakone. If you have less time, consider skipping the day trips to Kamakura and Nikko.