The 10 best places to go on day trips from Osaka or Kyoto would be the following: Nara, Kobe, Hiroshima, Miyajima, Kurashiki, Naoshima island, Nagoya, Mt. Koya, Kanazawa, Lake Biwa. If you have the JR Rail Pass, a lot of these places are accessible using the JR lines for all or part of the trip.
Nara is very popular to go to after visiting Kyoto, and Kobe is only a quick Shinkansen ride away. You can see the beautiful night view of Kobe city from atop the ropeway of Mt. Rokko, and be back at your hotel in Osaka or Kyoto before midnight. Or if you like art, go for a ferry ride on the Seto Inland Sea to Naoshima island which is full of art exhibits.
There's a lot you can see without having to move your base of operations so go enjoy Japan to its fullest!