Part 1:  First a short train cable car ride to get partway up into the mountain, where you can stop at the small but scenic Gora Park:

Part 2:  Then a cable car ride taking you over the steaming hot sulfuric springs of Owakudani, an area around a crater created by the last Mt. Hakone volcano about 3000 years ago. 

You can get off the cable car at Owakudani and walk fairly close to some of the hot springs.  And of course, you can purchase eggs that have been boiled in the natural hot springs.  The sulfuric acid turns the shell black (still taste like a freshly boiled egg inside, though); eating these eggs is said to prolong one’s life by seven years -- and for only 500 yen (~$5) for five of them!

Part 3:  A scenic boat ride across Lake Ashi:

Part 4:  A rather crowded bus ride back to the entrance station...not very picturesque...

There are a lot of places you can get out to take walks, in addition to museums and other interesting activities, which we missed in order to get around the circuit within the day.  Hakone is also just one part of the entire national park; it seems to be more popular for its resorts and onsen hot spring baths (for people -- not the ones the eggs are boiled in!) but it was enjoyable to see.  Plus, I ate three eggs (21 years worth), saw Hello Kitty in an eggshell, and added extra value to our JR Rail Pass.