My dds (14 & 11) really enjoy a museum, art gallery or gardens (like Kew or Wakehurst). Never done stately homes or NT as I dont drive, they are tricky to reach on public transport. TBH they just love a day out, we always make the best of it if it's not as good as we hoped!
I dont remember ever going to museums/art galleries or gardens as a child so all the new experiences are fun for me too. But I'm happy to be silly and get involved in putting on dressing up, building, the interactive games!
Shopping is my dd2s worst nightmare, she will cope with a market if there's cake or something she can buy. Dd1 loves mooching around the shops! Dd2 loves a park or beach, dd1 isn't keen on being outside for long periods of time, especially if it's hot!
But what I say to my 2 is, you have to try things before you decide if you like it or not. We went to the ballet, dd1 hated it and fell asleep, dd2 loves it and we've been every Christmas since. dd2 didn't enjoy local free festivals and gigs, so me and dd1 do that. They both loved the theatre even though they weren't sure before going.