First of all, there is no reason to switch hotels between Osaka and Kyoto. They are no more than an hour apart by train. Your list of cities tells me nothing about what you plan to see and do there, or if you have even researched this.
Second, it depends on whether you are more interested in modern, urban Japan or traditional Japan. If you are more interested in traditional Japan, spend more time in Kyoto and take perhaps one day trip to Osaka. For traditional Japan, definitely include a day trip from Kyoto to Nara (about 30 minutes by train). From Tokyo, Kamakura, home of the Great Buddha that seems to be a symbol of Japan in tourist-oriented materials, is only an hour by train and has other historic and artistic attractions as well.
First time tourists are often interested in seeing Mt. Fuji. As LapLap has told you, you can't climb it in April, and your chances of seeing it, even from Hakone, are maybe 50-50 in April, since it quickly disappears behind clouds if the air is humid. If the humidity is low, you may be able to see Mt. Fuji from one of the tall buildings in Tokyo that has an observation deck.
I would suggest buying two guidebooks, one for the Tokyo area and one for the Osaka-Kyoto area and figuring out what you want to see and do. Then, as LapLap suggests, plot these things out on maps, preferably a bilingual version of Google maps. Print out the maps for each area and carry them with you (or have them bookmarked on a phone or tablet). If you get lost, show your map to a passerby. In the large cities, your chances of finding a resident foreigner are pretty good, so such people can be an additional source of information.