A few days in Japan before going to Taiwan

Hi everyone !

I will spend 18 days in Japan and Taiwan with my girlfriend in April. We will arrive in Tokyo on Friday, April 11, and have our return flight from Taipei to Europe on Sunday, April 26. It will be our first time in Asia. We really want to spend most of our time in Taiwan but as our flight arrives in Tokyo, it's also an opportunity to spend time in Japan.

But early April is peak tourist season so for example I don't think we will enjoy Kyoto if we go there. We will definitely spend a couple of days in Tokyo, something like 3 or 4 full days.

Do you recommend any spot/city near Tokyo we can go for a day (or 2) easily ? Do you think we should spend more days in Japan ? Or the crowds of tourists prevent you from enjoying medium-sized tourist towns or spots such as Mount Fuji (so it may be a good idea to go to Taiwan after 3-4 days in Tokyo) ?

Thanks!