Going to Hakone by car is a bad idea unless you are familiar with driving on the left and really want to experience the often horrendous traffic on the way out of Tokyo. I was once on a bus to Tsukuba (since replaced by a train) that took over an hour to get out of Tokyo at 10AM on a weekday morning. It was stop-and-go until we reached the outer suburbs. Tsukuba is only 50 kilometers from Tokyo, and over half the trip was struggling through Tokyo traffic.

I was also once on a bus that took 3 hours to make it the approximately 50 miles from Narita Airport to Yokohama.

Long-term residents may find it advantageous to rent a car in certain circumstances, but the train system is so fast, efficient, and comprehensive that it seems like a waste for a short-term tourist to rent a car.

You have two options for going to Hakone by train: 1) Shinkansen from Tokyo Station to Odawara and then buy the Hakone Free Pass, which gives you access to the train, buses, funicular, the boat and aerial tramway, 2) The "Romance Car" train from Odakyu-Shinjuku (adjacent to JR Shinjuku) with the Hakone Free Pass.

If you have a reservation at an inn that is not on the standard tourist route, you can always find a bus or even a cab (I've seen cabs in the most out-of-the-way places) to take you there. Print out the inn's Japanese-language web page to show to the driver.