Do I need a 7 day or 14 day pass?

Hi there!

Using a 7 day JR Pass for your visit to Kyoto/Nara/Osaka is certainly a good option. The return Tokyo - Kyoto alone pays for the JR Pass and you can also use it to travel between Kyoto, Nara and Osaka (although some local transport methods such as the bus/metro) are not covered, however the JR network is very extensive and will get you to most places in Osaka.

As for the 14 day JR Pass, the difference in price with the 7 day JR Pass is 16,800 yen, in other words you would have to do at least this amount of travel with it for a 14 day JR Pass to be worth it. I made a small list of ticket prices to see if it would pay of.

Narita Airport - Tokyo ¥ 3,140
Tokyo - Nikko (return) ¥ 11,260
Tokyo - Kawaguchiko (return) ¥ 2,900 in savings
Tokyo - Narita Airport ¥ 3,140

For a total of ¥ 20,440 or ¥ 3,640 in savings so a 14 day JR Pass would pay of. There's a couple of things to note here though. First I used to route Tokyo - Nikko with the Shinkansen, you can also travel there by local trains - which is about half the price but takes about 1H longer and is much less comfortable.

Also please note that part of the route to Kawaguchiko is not covered by the JR Pass because the last part is traveled on the privately owned Fujikyu line after Otsuki station and requires an additional 1,110yen fare/one way. However there are still savings to be had with the JR Pass since you can use it to travel to Otsuki station, resulting in ¥ 2,900 savings.

Lastly, the JR Pass can be used to travel around Tokyo as well on the JR Network, which serves most area's. Given the last, I do think that a 14 day JR Pass is worth the upgrade, although it may be a little but cheaper if you use a 7 day JR Pass + local travel to Nikko.

I hope this helps!
Daniel