Tbh unless you're upper class you'll never truly be at home there. My dad comes from a mining family, and was a prison officer in his adult life. My mum worked in retail. We did lots of middle class things in the sense of holidays to nice places, eat out, (cruises, Lake Ghada etc) and had the money of middle class people, but my parent's values have remained working class. We sound working class. My dad is a die hard union man. We hate Tories and everyone where I live still loathes the likes of Margaret Thatcher. I'm from a poor Midlands town, so I'm posh for my area, but I had a real shock to the system when I went to uni.
Everyone sounded posh. Lots of people had private education. I was an A* student, but somehow felt stupid and put of place. People were into things like art, which is seen as pointless where I'm from. My parents thought art and etc was a waste of time. I'd never seen a play other than with school. I didn't have much cultural capital. I was equally as intelligent, but felt less educated, if that makes sense. I never even realised I had an accent, and then I went to uni and realised my Derbyshire accent is actually very broad lol.
People talked about gap years and month long trips to Japan like it was normal. My parents would never have funded something like that. They could have afforded it, but would have found it pointless as it's not for education or with the purpose of getting a job.
I will get along with whoever, but honestly, I feel a lot more at home in a Spoons with the truly working class (as these are the people I'm used to due to the area I grew up in) than I would sat with an Eton like crowd who have never been in a supermarket or been on holiday to Spain.
It's ridiculous to pretend it doesn't matter when we live in an exceptionally classist country driven by socioeconomic background. It blatantly does matter. This country is divided by class in the same way America is divided by race.
But anyway. Give your kids a mix of experiences. Take them to plays, help them be well read, take them to museums, save and help them travel, teach them good manners, take them to formal dinner type settings. But equally take them to pubs, let them do typical working class hobbies like football, go on a beach holiday. Then you'll have kids who will hopefully fit in wherever but won't be stuck up. But tbh your class is more than activities- it's about your values and where you've come from. In the same way you will never quite grasp the nuances of any culture unless you've grown up there, you can't truly understand what it is to be upper class unless you are. The same goes for upper class people understanding the working class too.