25 days in Japan - will travel to Tokyo - Osaka - Nara - Kyoto - Tokyo

Hi, my family and I will be visiting Japan on the 18 Jan 2018 arriving at the Narita Airport on the same day. We are going to stay at Mejiro, Toshima-ku, Tokyo. For the first week, we're planning to visit Osaka, Nara and Kyoto then back to Tokyo. We also want to see Mt. Fuji and go to Yokohama. Then, we want to spend the rest of our days in Tokyo.

I would appreciate for any suggestions/help you could give me regarding of which JR passes we should get or is it worth getting one? How about some good places or tourist spot that we can visit in Tokyo, Osaka and Kyoto? We're going back home and leaving Tokyo (Narita Airport) on 12 Feb 2018.

Thank you for your help.