We went to Japan 5.5 years back. We mostly planned it ourselves and extensively used the JR pass for travels. Things we did... (In order)
Nara (1 evening+day) - the standard touristy stuff in the morning and in evening we attended a light festival that was very very pretty.
Koyasan (1 evening+day) - we didn't know what to expect and TBH didn't expect much but it ended up being one of our favourite points in the trip. You basically stay at a Buddhist monastery on top of a mountain. You get to attend their daily ritual if you wish in the morning and you dive at the monastery with the monks. It can all be a bit formal but fun, relaxing and eye opening at the same time.
Kyoto (1 evening + 2 days) - surprisingly our least favourite bit. The touristy bits are very nice and pretty. The outskirts are quite nice too but the city itself is a bit grey. We also attended the daimonji festival here, it's not for the faint of hearts but great fun.
Hiroshima (2 days) - very pretty place and an interesting place to visit. The history of this place and museum hits your feels quite hard. The museum especially is very hard to get through but you'll be glad you did it none the less. We also did a day in the miyajima island just off Hiroshima.
Mt. Fuji (2 days) - hiked to the summit. Enough said
Tokyo (3 days) - lots to do here. We went shopping, went on Odiba which is a man made island (saw Asimo here), went up the world trade center, the isn't shortage of stuff to do here..... Just pick your favourite in the time you have left (that's what we did otherwise you'll end up spending all the time here lol)
I have never felt as safe anywhere else as I felt in Japan inc. London (ok don't laugh, London is kinda safe!). For the main issue was finding vegetarian food but if you don't have this issue then finding good food is not hard at all. My friends tell me even the McDonald's is actually decent in Japan. Their standards is definitely a level above in terms of service and quality.