Hi OP
I live in Japan, on 30 minutes from downtown Tokyo with Japanese DH.
Regular poster but used namechange as this is outing.
Kyoto: gorgeous place with amazing food and atmosphere. Arashima forest is beautiful but very touristic. Worth a visit though. If you have seen the movie Memoirs of a Geisha, visit Gion too. It instantly transfers you back in time!
Hiroshima: Mijayima is very beautiful and definitely worth a visit. The island has a beautiful shrine and lovely shops. And very cute deers wandering around the streets!
The monument for the atomic bomb victims is very impressive and worth a visit.
Don’t forget to pick up some of the momiji manju cookies, they are a regional specialty from there and delicious!
Tokyo: since you have teens, it’s a nice idea to visit the gorgeous Meiji shrine and explore Harajuku with them afterwards (Sangubashi station or Harajuku station). Shibuya is also real nice for teens and if you like sushi, the Standing Sushi Bar there is highly recommended!
I also love Asakusa for the retro atmosphere (again very much tourists though). Ueno is also popular with tourists but not very nice IMO, very shabby.
Re transportation, as others have said get a Japan Railcard. Shinkansen trains are very expensive and I believe even a return ticket from Tokyo to Kyoto already costs more then the rail card!
Order them before you leave since they are only available outside Japan and make sure you take your passport on the train with you as they want to see that as a proof you are using them as a non - resident.
Re going from Haneda to Tokyo, I always take a bus (Friendly Airport Limousine Bus), from Narita you can take the Narita Express to Shinjuku station at least, I think also to Tokyo Central Station.
Re things to do around Tokyo: consider Hakone as well, gorgeous village on a steep mountain hill surrounded by nature and famous for onsens. Go on the ship at the lake to see the beautiful nature around the lake and if you are lucky and it’s not foggy, Fuji san.
Kamakura, an old city near the coast is also worth a visit. Beautiful shrines and temples, devoted to Samurai families. Very very impressive.
Enjoy Japan and if you need any more advice, please PM me.