Good for you for thinking about this. “Be yourself” is all very well but having knowledge and experiences which underpin your confidence in professional and social situations is valuable in life.
Given their current ages, Aquila magazine, or similar - National Geographic junior etc. Look for back copies of Aquila on eBay as there was a golden period several years ago and the content doesn’t really date too much.
Learn an instrument, and join the orchestra (most local authorities will fund the provision so it is not free but is heavily subsidised). This is a social world that tilts UK traditional middle/educated class and educated/aspirational families from other ethnic backgrounds too. Even if you children don’t continue to play long term, they will be exposed to the “world” of music and kids outside their school.
A Saturday broadsheet in print. Subscription to The Economist when they are around year 10. What they read matters. Broadsheets aren’t what they used to be.
Many people who have been brought up in middle class environments will have had Radio 4 constantly on in the background of their lives… it gives a different sort of lifetime cultural capital to going to museums and galleries etc. I had to endure the Today programme every weekday morning for my entire life until I left home. It was boring and I didn’t like it but it built an awareness of the world, and what educated and/or powerful people think and how they communicate and discuss things, that you do not get from reading news online. Similarly desert island discs. I think radio like this can be superior to TV news, although that can be good too at times. Also Radio 4 podcasts like In Our Time and “Add to playlist” (which is a joy!).
Radio 3; even if they don’t always understand (or care!) what the presenters are talking about or like the music it is immersion in a (privileged) world beyond their own. Stick it on in the car. People on Radio 3 are so charming and polite to each other. I find it a balm, but cheerfully admit I will switch over to pop channels when the music is “hard-going non-melodic unharmonious” stuff or the more dramatic end of opera! Exploring media beyond pop culture is important. Middle class kids endure lots of stuff that is “character building” - they don’t enjoy all of it, lol.
I strongly agree that sports can also be an effective way to cross class barriers.
Getting used to being outside your comfort zone and doing things which are not your usual preference might be initially unpalatable but then often you will “get” it or build familiarity or start to enjoy it, and suddenly something new and worthwhile opens up. This builds resilience and perspective. These are the sorts of things that fee-paying schools instil and that’s why kids coming out of them often have that “money can’t buy” inner confidence (but obviously money has bought it!). They have the confidence to be interested rather than intimidated when facing unfamiliar situations or topics.
Finally, you can lead by example and this is really powerful. By trying something new, by being open minded and curious - which even by your OP you are showing that you are. 😊
Go to an evening class/club, a book group, take up online learning, dip your toe in language learning on Duolingo… I know time is challenging with kids so some of this might be on the to do list for a while but it’s worth thinking about how to squeeze a bit of your own self actualisation in…