I think I've learned as much about Italy when relaxing on a beach (one not on the main tourist track) or lingering in a beachfront restaurant, surrounded by Italians, than when trekking through a museum checking off "must see" works of art. But that's just my own choice...
If you are against relaxing all the time, which I am not and I don't think you are based on the title of your post, one of the loveliest places to visit on Ischia are the Giardini la Mortella, the enchanting gardens planned and planted by Susanna Walton:
https://www.lamortella.org/en/
A town on the Amalfi Coast that I adore, and that is not included on most tours, is Cetara, towards the end of the coast near Vietri.
Far less English-seaking tourists than the more famous towns, this is what that Coast might have been like fifty-seventy years ago. We never go to that area without paying a visit, and having a meal art one of the fantastic restaurants lining the Main Street leading down to the sea. But there is only one hotel of note and it's invariably booked far in advance, as I've learned the hard way. I don't think any Amalfi Coast town has better restaurants than Cetara.
You can't go wrong with Ischia!!