I am bringing my DCs up in a similar way to how I was brought up. I've never thought of it as cultural capital, but aged 8 and 10 they:
Have visited art galleries and museums (history, science, ...) and historical sites in several cities in 3 different countries (soon to be 5 by September).
Are bilingual - this is key as we want them to experience the culture of both our countries
Have been to the theatre several times in both countries
Read books in both languages
Are learning about history and traditions in both our countries and others around the world (through discussions and TV programs)
Do a couple of different sports each. We have a piano and they play around on it but as they are dyspraxic and haven't shown a lot of interest I haven't got them lessons yet.
Have travelled to several different countries - both for the nature and the towns.
Have done many walks in forests and mountains and moorlands etc.
Have visited zoos, aquariums and watched lots of nature programs.
Its all been gently introduced as things to try and to do and then we've followed their interests (history and animals). Both my DSes have ADHD so we have always needed to be very active with them at the weekend and in the school holidays so it all adds up. Plus we live in Paris with many museums on our doorstep and my parents are very enthusiastic at taking them to new places.