We brought our DC up in zone 2. We now live just outside.
When the DC were young, we did a lot of museum dipping- it was close enough to just go and see the bits they really wanted to see. Also the galleries.
DH is an opera buff so we do go to the RoH every six to eight weeks. When he worked commercially it often was combined with clients and a good restaurant or a good restaurant on its own. We used to go to more concerts, but do this less now. DD and bf are the family prommers but both very musical.
My hairdresser is in SW London and every seven weeks I either go shopping, usually Peter Jones, sometimes Marylebone, or to an exhibition that interests me. I make a day of it.
We occasionally socialise in Central London, usually at a Club.
Also when the DC were younger much activity focused on Sports clubs: The Hurlingham and Rosslyn Park in particular.
One of the plusses is that both DC can navigate London on the backs of their hands and I think a London child grows up a little more streetwise.
I loved living in London and miss it greatly. DH wanted the big house and garden. I would go back in a heartbeat.
Even on a wet day London is about yellow signs on cabs reflecting in puddles, bright lights, bustle and good shops. There is also beautiful architecture and you can never get lost if you follow the river. The lights on the river as you look over to South Bank and across the bridges are a tonic for the soul. And the parks, don't forget the parks - the bandstand in St James's, a pretty walk and gentle snog with a lover in one's youth 😉.