8 day, 7 night trip

My friend and I are planning our first trip to Japan in October. Our flights are already confirmed. Does this look like a good plan?

day 1: arrive Osaka KIX
day 2: Osaka
day 3: Osaka - Nara - Osaka
day 4: Osaka -Kyoto
day 5: Kyoto - Hakone (Lake Ashi, onsen, view Mt Fuji)
day 6: Hakone - Tokyo
day 7: Tokyo
day 8: depart Tokyo (Narita)

Sounds fine. You don't have to move accommodation to Kyoto; you can visit it as a day trip from Osaka.

day 1: arrive Osaka KIX
day 2: Osaka
day 3: Osaka - Nara - Kyoto
day 4: Kyoto
day 5: Kyoto - Tokyo
day 6: Tokyo-Hakone - Tokyo
day 7: Tokyo
day 8: depart Tokyo (Narita)

I disagree with Mudo - head straight to Hakone from Kyoto.

thanks for the replies. What do you recommend for trains & public transportation between the cities and within each city?

Hi
Depends on what you are going to see

From Kyoto to Hakone you will want to take the JR Train and then bus to the location(s) you wish to visit
Stay overnight?
bus back to the station and take JR train to Tokyo

Kyoto - Depends on where you are going, but there is a good bus system for central, northern and eastern areas and JR train to Arashiyama (western)

Tokyo - trains/subways everywhere in Tokyo. Take the JR train to Narita airport unless you are staying at Ueno. Take the skyline to airport if you are staying at Ueno

Hope this helps