Hi there,
I recommend using the following types of transportation.
Narita - Tokyo, use the Narita Express.
Tokyo - Hakone, use the Shinkansen bullet train from Tokyo station to Odawara. From Odawara local transport can be used throughout the Hakone area.
From Odawara, Shinkansen trains travel directly to Kyoto and Osaka, you can transfer from Kyoto on a local line to Nara.
As for your last day, I would recommend staying near Tokyo or Narita, this way you are sure to be in time for your home flight.
Using a 7 day JR Pass here would help you make good savings, compared to purchasing normal tickets.
Hope this helps,
Daniel