I'm just back from a couple of weeks in Japan. Just to add my two cents on some of the points others have raised, with the IC cards like Suica, if you have an Apple phone you can choose the regular or Welcome (tourist) versions, whereas with a non-Japanese Android phone you can only get the Welcome version through the app (AFAIK). We just bought physical regular cards (ICOCA, as we landed in Osaka, and Suica is the Tokyo area one, though they generally work throughout the country).
We did a day trip to Mount Fuji from Tokyo on our only organised tour. It was a bit meh but I think hard to do the various places of interest around there in one day otherwise. We were lucky to get a clear view of the mountain in the afternoon, as it's often not possible at this time of year, but I've been to Japan before, in April, and the snow topped version is so much more memorable than the current snowless one.
We also went to Kamakura from Tokyo. It's pretty busy with tourists, so I'd say go early. We enjoyed the ride on the little local Enoden (Enoshima) line from Kamakura. Go in the front carriage right behind the driver for an amazing view.
The temples in most places open early (like 6am) so for the popular ones it can be much nicer going early. Especially recommend doing that in Kyoto at Kiyomizu-dera as the lovely surrounding streets are also quiet at that time. All temples take cash only (if they charge).
We had a couple of days in Kanazawa and I enjoyed it. It's a bit cheaper for accommodation and food, and good for seafood at the Omicho market. Try the restaurants on the second floor.
In Kyoto we had a three-person hotel room at the Good Nature Hotel, which was very nice, though better if it's just two people as you basically sacrifice a sofa. It's normally pricy but we got a good deal.
We had apartments in Osaka and Tokyo. The Osaka one was at Ocean Tsutenkaku. We had the family apartment which was very spacious for the price. You need to be a bit tech savvy for it, as it's all digital (entry, check-in, doors) but good value.
Edit to add, used an e-sim from Airalo and it worked great, including adding a data topup.