All the places you've listed: Kamakura, Nikko, Hakone, are great and recommended to visit from Tokyo :)
Especially in Hakone or Nikko (or Kinugawa, it's about 10 minutes drive from Nikko), I would like to recommend to stay in an Onsen Ryokan at least a night. Experiencing a stay in Onsen Ryokan would be memorable with beautiful Japanese style room, futon, yukata as pajamas. Order dinner and breakfast of the ryokan. The dishes are amazing, created with local ingredients, and so many plates to make your stomach satisfied. A soak in onsen need no words. Experience both inner and outer bath.
When you go to Hakone, drop off at Odawara and visit Odawara castle. Seeing a Japanese castle would be a nice experience.
If you would like to visit more cities, add Kawagoe. It's a small city in Saitama Prefecture with historical buildings.