Hakone is about 1.5 hours by train from Tokyo. You should purchase the Hakone Freepass (about $55), which gives you a round-trip ticket on the express train from Shinjuku Station in Tokyo to Odawara. It also includes the mountain tram, cable car, ropeway, boat, and bus. We took a circular journey around the area using all the above mentioned forms of transportation. It was fun and we saw a lot of lovely scenery. You have to leave Tokyo by 7am if you want to do the trip in one day.
When you get to Odawara, you then board the Hakone Tozan Electric train, which takes you to Hakone-Yumoto. You then board the mountain tram for Gora. It zigzags its way up the mountain. At times, it goes on a siding and the conductor and engineer change places as they get on another track and continue up the mountain. It's a lovely scenic ride. You can get off the tram in Chokoku-no-Mori if you wish to visit the famous Hakone Open-Air Museum. There are 700 sculptures on display both outdoors and in several buildings. Then get back on tram and go to Gora. Here is where a cable car takes you further up the mountain until you reach Souzan.
Then you board a ropeway and travel in a gondola. The first stop is at Owakudani, where you'll see sulfurous steam escaping from fissures in the rock. There is a nature trail there. There are also restaurants and souvenir shops. Here you get your first view of Mt. Fuji. When you reboard the gondola, you will get spectacular views of Mt. Fuji as you travel down to Lake Ashi. When you get there, you board a boat for the ride across the lake to Moto-Hakone. Nearby the town is an ancient cedar avenue that you can walk through. Then a bus will take you back to Hakone-Yumato, where you can take the train back to Tokyo.