The Hakone area is a nature area in the mountains, also famous for the large amount of spas (hot springs). Good place for hiking, with view of Mt Fuji, and various other activities, such as ropeways, open air museum, and boat tours on the lakes.
While it is possible to both go to Hakone with train and go around with bus and train, the best way to go around is to use a car (or join a tour). Many of the nicer nature paths and shrines etc are not directly accessible by train.
How to get there
From Tokyo, start with transferring to Odawara. The fastest is to take Shinkansen , though you have to expect the double cost. If not taking Shinkansen, Tokaido line to Odawara is recommended.
From Odawara, take Odakyu line (or Hakone Tozan Railway) to Hakone Yumoto.
From Tokyo, go to Shinjuku (multiple options). From Shinjuku, take the Odakyu Ltd. Express to Hakone Yumoto.