I am very curious about people who enjoy being "immersed in local culture" when they travel. I have traveled quite a bit, both for work and for pleasure, and I have never experienced this feeling (but would like to).
How long do you have to stay in a place for this to happen? What activities do you do that enhance that experience? Certainly you must be doing something different than going to museums and seeing tourist sites....so what is it?
The times that come to mind are those where I've met and befriended some local people, spent time in their homes and in their lives and got to know them. Some times it takes days, other times I've been in a place for a lot longer and have not had the opportunity. Language barrier makes it harder, but that's probably more so for me, being shy and afraid to make mistakes (working on that). It's also a lot easier if you know people who know people already.
Went to Bosnia with a boyfriend at the time who had friends and family there. Many of them didn't speak English and I don't know Bosnian, but I still got to live in their homes and live their lives for about 6 weeks, to experience cultural differences. Eat where and what they eat, drink where they drink, socialize, party, go to their cottage for the weekend, etc.
The only example I have that compares the difference is in the Cook Islands. I went there twice. Once on my way to Australia, once on the way back. On the way there I did all the tourist stuff, including a Maori show. When I stopped there again on the way back I met a few locals and hung out with them. Visited their homes, let them show me their island from their perspective. One night we ended up going to the same 'show', but watched from backstage, instead of with the tourists.
So to answer the questions:
How long do you have to stay in a place for this to happen?
The longer the better, but if you are lucky, only days. It's also probably dependent on where you are. How touristy the location is and how friendly the locals are.
What activities do you do that enhance that experience?
Meet the locals.
Certainly you must be doing something different than going to museums and seeing tourist sites....so what is it?
Go to where the locals go, not where the tourists go.