Tokyo Hakone Kyoto

Hi, we are first time visitors to Japan and I would like to know a few things:
1. Is it easy to tour Japan on our own inlcuding a 1 day trip to Hakone and 2 days in Kyoto?
2. Given out itinerary, which are in Tokyo is the best place to stay? roponggi, shinjuku or shibuya?

Thanks in advance for your help.

1. Is it easy to tour Japan on our own inlcuding a 1 day trip to Hakone and 2 days in Kyoto?

Yes. It takes some effort, but if you read the guide books or websites about how to do it, it is pretty easy. Here is a good starting point for Hakone:
http://www.japan-guide.com/e/e5200.html

And for Kyoto:
http://www.japan-guide.com/e/e2158.html

2. Given out itinerary, which are in Tokyo is the best place to stay? roponggi, shinjuku or shibuya?

Hakone is most easily accessed from the Shinjuku or Tokyo Station area. Kyoto is most conveniently accessed from the Tokyo Station or Shinagawa Station area. But I assume you will also spend some days in Tokyo in which case I think you should also consider each district's character when deciding. Here is an overview:
http://www.japan-guide.com/hotels/tokyo.html

Thank you Uji for your helpful reply. We plan to do our itinerary as follows:
Day 1 & 2 - Tokyo ( we will arrive at night on Day 1 so thats practically a lost day)
Day 3 - Hakone/Mt. Fuji then proceed to Kyoto for overnight
Day 4 - Kyoto/Nara / Kyoto
Day 5 - Kyoto/Tokyo
Day 6 - Day 8 - Tokyo
Can you recommend a good hotel near the station in Shinjuku and Kyoto?
We hope to arrive during the Cherry Blossom season (we arrive April 6) and hopefully we the picked the right travel date.

Can you recommend a good hotel near the station in Shinjuku and Kyoto?

My favorites are the Hotel Century Southern Tower in Shinjuku (thanks to its highly convenient location) and the Granvia Kyoto (again because of its very convenient location). Note that Kyoto is very busy during the cherry blossom season, and you better make the hotel reservations as soon as possible. Especially Saturdays (April 6) tend to be extremely busy. You may want to delay your visit to Kyoto by a day for more hotel choices.