Modes of transport in the Hakone region are many and varied. Your options include:
The scenic Hakone-Tozan Line mountain railway from Odawara to Gora via Hakone-Yumoto
The Hakone-Tozan Cablecar up the mountainside from Gora to Sounzan
The Hakone Ropeway from Sounzan down to Togendai on Lake Ashinoko via the boiling sulphur pits of Owakudani
The Hakone Sightseeing Ships, decked out like Disneyland versions of pirate ships, sailing across the lake from Togendai to Moto-Hakone and Hakone-machi
And positively dull in comparison, the Odakyu Bus back to Hakone-Yumoto or Odawara
Portions of the above circuit may close for a short period of time in the winter for maintenance, but shuttle buses replace the closed services, so this won't ruin your trip.
Most people opt for the Odakyu Hakone Free Pass (http://www.odakyu.jp/engl...), which includes a return trip from Shinjuku and allows unlimited use of all of the above forms of transport for several days. In addition, pass holders can receive discounts at many hot springs, museums, restaurants, and other locations by showing their pass.
The 2-day Free Pass costs ¥5000 from Shinjuku and ¥3900 from Odawara. A 3-day pass can be purchased for an extra ¥500. Additionally, if you have a Free Pass from Shinjuku you can use the Hakone Limited Express train by paying a surcharge of ¥870 each way. Holders of the Japan Rail Pass and JR East Rail Pass should travel to Odawara station by JR using the methods described earlier, and pay for the Free Pass from Odawara.
