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The area has some nice hiking trails. The visitors' center in Togendai has a model of the area including many of the hikes around the Lake Ashi. For example there is a hike from Togendai to Owakudani and from there to Mt. Komagatake passing the peaks of Mt. Kanmurigatake and Mt. Kami. From Mt. Komagatake take the cable car down and the bus back to Togendai (beware: last cable car leaves Mt. Komagatake at 16:50). The hike takes between 2½ and 3 hours and covers quite a difference in altitude. Sturdy shoes are definitely helpful but not necessary.
大涌谷 Ōwakudani
Volcanic hot spot full of sulphurous springs.
Nestled on the south shore of the lake, close to Moto-Hakone, is a picturesque Shinto shrine with torii gates in water.
Ashinoko
Beautiful views of Mount Fuji but only on a clear day. As many tourists have found out, a visit to Lake Ashinoko does not guarantee a view of the mountain. The lake is crisscrossed by cartoonishly decorated "pirate ships".
箱根 彫刻の森美術館, Hakone Choukoku no Mori Bijutsukan
One of the top attractions in Hakone. Displays a wide variety of sculptures and artwork from celebrated Western and Japanese artists within a beautiful parkland setting. Also includes a dedicated Picasso pavilion (paintings and pottery).
ポーラ美術館, Pola Bijutsukan
The Pola Museum of Art's permanent collection is centered on works gathered over a 40-year period by Suzuki Tsuneshi, the son of the founder of Pola, a Japanese health and beauty company. The collection of around 10,000 items spans a wide range of genres including Western painting, Japanese Western-style painting, Japanese-style painting, prints, sculpture, Oriental ceramics, Japanese modern and contemporary ceramics, glasswork, and cosmetic utensils and other objects. Amongst Western masterpieces, the collection includes impressionist works from Monet, Manet, Renoir and Degas, as well as from Picasso.
箱根写真美術館 or Hakone Shiashin Bijutsukan
Small museum with nice pictures of Hakone and Mount Fuji.