I think it’s more about your outlook as a family than about specific activities. If you are keen to learn about the world around you, your children will pick up on this attitude.
Discuss the news or sport or music or history or philosophy or whatever comes up. Perhaps take out a subscription to one of the children’s weekly news magazines, and take the time to read some of it with your child, and debate the week’s topic etc.
If interest is sparked, follow it up. For example, our DD was interested in Richard III after watching or reading Horrible Histories, so we had a day trip out to Leicester to see the exhibition, and to the Bosworth Field battle site.
In terms of sport, music, languages etc, I do think you need to be led by the child. We’re not a sporty family, and we didn’t feel it was right to force DD to do a sport when she wasn't keen, though we’re musical and she now learns two instruments. When she didn’t enjoy Rainbows, we offered her some alternative options and she chose to learn a language, and has been going to a small group class ever since. We wouldn’t have forced this upon her though: it genuinely followed her interest after a family holiday abroad.
I don’t think there is a perfect formula for enrichment. I’m also not convinced that private schools offer the perfect balance (though DH and I were state educated, so maybe we don’t know what we’re missing.)