OP I come from a real mix of class backgrounds both from parents and then i married 'well' (not by design!!) so i've been watching this play out my whole life and thought about it a lot (esp as i grew up outside UK for a while so had an outsiders view too)
1 - confidence and ability to look people in the eye is crucial. my DC8 is in public speaking/drama classes to help with confidence - i want them to be more confident than i was through my school and uni years
2 - cultural influences - we spent childhood in museums and art galleries and accepted this was the norm. My parents liked it because it was free (we had little spending money) but it taught us to respect perspectives and learning.
3 - Do this stuff WITH your kids, don't just tell them they should be doing it. This ultimately teaches them that its an important part of being an interesting adult.
4 - Holidays - there are absolutely LOADS of historic things to see in the UK very cheaply (Roman, medieval, iron age forts etc) Helps if you like history obviously. To this day I can't go on holiday here or abroad without always visiting an important site or museum if there is one in the area. It makes me sad that so many people go abroad to visit shopping malls to go to Zara & H&M (I often like to do this too, but not until i've exhausted the other stuff)
5 - Teach them manners - boys AND girls should be taught to hold doors open for people and offer seats on the tube, etc. And they need to know how to hold their cutlery - if only so they feel comfortable on dates and at business meals out when they are older!