Hakone is one of the most popular tourist resorts in Japan—Japanese tourists as well as foreign. It is a beautiful region of forested mountains with a large lake and many hot springs resorts. In the summer it offers a cool respite from the heat of Tokyo. Visit in the fall for fiery maple leaves or the spring for cherry blossoms and azaleas. In the winter you can enjoy a excellent view of snow-capped Mt. Fuji.
There is so much to explore here that I recommend picking an area and concentrating on that. These are some areas and their highlights:
The gateway to the spa district
Parks, museums and hot springs in an ancient volcanic crater. One of my favorites here is the Open Air Museum
Sengokuhara/Northern Lake Ashi
The place for sports enthusiasts. Here you can enjoy golf, boating and fishing
Moto-Hakone / Hakone-Machi
Located at the southern end of Lake Ashi this area has everything from culture, to shopping, to recreation.
Hakone is a 1 hour, 30 minute train ride from Shinjuku station. I recommend getting the Hakone Free Pass with optional train fare from Shinjuku. This pass also covers unlimited use of Odakyu affiliated buses, trains, boats, cable cars and ropeways within the Hakone area. It provides discounted admission to many museums and attractions along the way too. Passes are 2-day or 3-day and can be purchased at Shinjuku station or stations in Hakone.