Japan is amazing. you'll love it. It's clean, safe, friendly, fascinating, stunningly beautiful with so many varied things to do.
I was glad we flew into Osaka and out of Tokyo as I wouldn't have fancied Tokyo with jet lag. Osaka is a much easier city to navigate if you are really tired. It has a pretty castle and really cool USA retro area full of games cafes and stunning architecture. The walk along the river with all the reflected coloured lights is amazing too.
Spend at least 3 nights, ideally 4 or 5 in Kyoto. There is SO much to see. Day trip to Nara, Golden and Silver Pavilions, incredible food markets, geisha district, the castle, Fushimi Inari etc.
We loved Arashiyama where you climb a hill to feed wild monkeys. There's also the golden pirate galleon boat trip and bamboo grove but the monkeys were the best bit imo. There's also a shop which sells pikachu shaped cakes and donuts - so cute.
Hakone is lovely with its onsen hot springs and gorgeous outdoor sculpture park as well as the cable car ride and stop off to the weird sulphur springs where they boil black eggs.
We did gorgeous hikes along the old Samurai trail - like walking into a Ghibli film (don't miss Ghibli museum in Tokyo if your children are fans - mine loved it)
My favourite moment was staying in a Buddhist nomastery on top of Mount Kobe. Everything about those two days felt like living in a fairy tale.
Tokyo is great - the crazy fashion in Harajuku, the shopping, the amazing museums, the sushi fish market and other amazing food experiences (the singing ice cream makers, the yakitori alley.