I have been to Hakone and I think Kawaguchiko is much nicer. Hakone is further away from Mt Fuji. One of the Fuji 5 lakes, Kawaguchiko is the most visited lake among the 5 but it was cloudy when we were there =( recommended places to visit and things to do include the Ukai music museum where the roses were in bloom (check the season) and you get to listen to a short recital by a Prague string quartet! The roses were so beautiful! My mother-in-law would have loved it! =D the natural living centre is where lavender will be in bloom in july and where you can see Mt Fuji (if it lets you). You can learn how to make blueberry jam (blueberries are grown near Mt Fuji) for JPY800 per person (min 2 persons) and take home your jam =)
For JPY2000, you get a retro sightseeing bus ticket (it’s a retro bus), a cruise ticket and a ropeway ticket. The bus ticket allows unlimited rides for two days. You can request for the tour info staff to forward your baggage to your hotel/ryokan at no charge =) so that you can do your sightseeing and check-in.
To get to kawaguchiko, we took the JR from Shinjuku to Otsuki then transferred to the Fujikyu limited express (non-JR), a very cute train with Mt Fuji cartoon painted on the outside. If you pay JPY100 extra, you get to sit at the wide-view or lounge seats =) Coming back, we took the Tozandensha, an old-style train with beautiful interior =D very very nice. Every station we stopped at, people were taking photos of our train hahaha.