Hi Lia,
Arrival at Narita Airport
There is a Airport limousine available that travels directly to Maihama and your Hotel. The price is ¥ 2400 and leaves about every hour. I would definitely recommend taking the transport option if you arrived all packed. You can find more information about it via this link.
You can also take the train there, this route is free with the JR Pass but does involve a couple of transfers, here is the full route.
Going to Hakone
Hakone is the area around Fuji and includes many smaller places, because if this it is a little hard to give you the exact route. Most people travel to Tokyo station and take the Bullet train from there to Odawara, this is the access station for Hakone. You can travel on from there to your accommodation with various local transport methods.
On to Kyoto, Nara and Osaka
The travel time to Kyoto from Hakone is 123 Minutes (roughly 2 hours) by Hikari Shinkansen (a certain class of bullet train). Hikari trains leave every 30 minutes during the day. For instance if you take the train at 18:08 from Odawara, then you arrive at 20:11 in Kyoto, giving you almost the full day in Hakone and enough time to check into your hotel in Kyoto.
Nara can be reached directly from Kyoto station in around 1 hour. The trip is very easy and you can enjoy a nice view of the Kansai area from the train. In Nara I would recommend a visit to the Tōdai-ji temple, this home houses the big Buddha and you will be sure to meet some of the free roaming deers.
You can't really miss a visit to Osaka if you come to Japan, Osaka can be reached in around 30minutes from directly from Kyoto station, so just get into the train and get out in Osaka. There really is a lot to see and do in Osaka I would recommend reading this site. One thing I certainly would recommend is to walk the streets of Shinsai Bashi, the area is great for buying all kinds of small things and I think it gives a great impression of Osaka.
I hope this helps,
Daniel