Seeking advice for a first mid July trip

Hello,

I am going with my girlfriend to japan from the 10 to 24 July (i know not the perfect season huh)
And we are really hesitating about our itinerary.
We know that we will spend at least 4 days in Kyoto and 6 in Tokyo.
We tought about doing a daytrip to Nara and to Kamakura
I have seen that the nature is very « green » so and i think about staying one night in Nikko and maybe hike the Nikko-Shirane (i really love the picture that i have seen of the hike)
But is it easily doable or even accessible without a car ? If so do you know any hike or landscape that looks like this ? (Just completly green montains with a lake if possible)
Also should we go to Osaka ? Or somwhere else ? We are really open to any suggestions especially anything that would be kinda romantic ahah
I think that doing only kyoto tokyo and nikko is good and adding something else would rush the things a little but the trip is soon and i want to know your opinion if we missed something ?

Thanks !!