If you are set on going, then I would rather concentrate the 1st trip to Tokyo, or Osaka and Kyoto. Rather than Tokyo + Kyoto. The reason being that to do your original plan, you are really best fly into one and out of the other. That can push costs up even further, not to mention a 1 way trip on the Shinkansen is like £100 per person.
If you are landing in Tokyo, pick Haneda, NOT Narita. Narita is about 90min train via the NEX Narita Express and cost like £30. So if the flight to Haneda is like £50 more...it's worth it. Haneda is in the city, it is connected by a monorail, close enough for a £20 taxi to the hotel.
If you are flying into Kansai Airport then it's about an hour to Kyoto from there.
As for getting train tickets, you can get it from the machines, which has English, or go to one of the counters, and they have English speaker there. Prepare to give yourself like 20mins though to get tickets to find the platform if you buy it on the day, and you won't be able to book a seat on the day. So you will need to go to the unreserved carriages. If you book ahead you can reserve a seat. Try to pack light, as LARGE suitcases are only allowed if you book a space, but overhead storage can fit a medium size suitcase. Don't be one of those people who put a suitcase in front of the seats...the locals will hate you.
Hotels right now is crazy, so just pick somewhere close to the main station in Kyoto. Kyoto transport is erm....crowded...well, the buses are, I hate Kyoto buses, I rather walk 2 miles than take the bus in the day time at least. So many tourists .... i am part of the problem so i walk. The trains and subway are not bad but they don't go everywhere in the city so some places, especially on the East side, where Kiyomizu Dera is are bus access only. 30mins on foot from the station...I just walk it. If they have iPhones then get the Suica app in the wallet, just top it up and tap tap tap on buses, trains all over the country. If not, then you need to get the travel Suica card...they have currently suspended selling the normal Suica card due to Chip shortages.
As for what to do....just temples and temples really. The place is full of history and old Japan, so if you like then it's amazing but also...a lot of tourists kinda ruins that a bit. There are other smaller less famous places that has the same feel, without the crowds.
In Tokyo, stay on any of the station on the Yamanote line, I suggest either west side (Shibuya, Shinjuku up to Ikebukuro) or East side, from Shimbashi up to Uneo, Akihabara. Avoid staying in the North or South area, not much main attractions around those. If you are on the Yamanote Line then everything wil be quicke to get to.
I went to Singapore for a week during the pandemic, but my flight was also like £500...so much better value. I guess you can go to Japan for a week but I feel like, unlike Singapore, you barely scratch the surface or even get used to the customs in that time. Not to mention right now it is like a sauna, i hope they like sweating...but at least they will get a chance to try Pocari Sweat.
