I understand that you have ten days in Japan, and you want to visit both Kyoto and Osaka.
The first three or four days you’d be staying in Kyoto, perhaps a traditional Japanese inn to add to the ambience, and would be visiting the city’s beautiful temples and shrines, old neighbourhoods and sites full of history. After this, the following five or six days would be open to visiting nearby towns and cities and getting to know more of the country: including to Osaka as well as Japan’s biggest castle, a short hike in the mountains, a town full of roaming deer in the streets, hot springs, and a sacred island. We can also include a visit to Hiroshima if you would like to get to know more about that side of Japanese history.
The remaining day is up in the air, because December has many festivals and I think it would be a good idea to try and fit one in during your stay. Festivals have many traditional events like dances or parades, sometimes music, and often include festival food which many love to try.
Of course, we can change anything that doesn’t fully appeal! You’d also have the chance to try the local specialities of all these places, and some of my favourite restaurants in Kyoto.