The most common way for foreigners to go study in China is to get a CSC scholarship. You won't live in somebody's home, but rather a dorm room. Still, you can get out and socialize with the Chinese all you want. I am sure they will be more than happy to help you with your Chinese if you help them with their English. The CSC scholarship award can cover your tuition, and books. You also get a small monthly stipend which is probably enough to live off of if you are frugal, but you should really bring as much money as you can if you want to have fun.
I believe it is also "illegal" to work on a student visa, yet I hear of people doing it all the time. Still, it's obviously a risk and you could be deported.
By the way, I haven't done this yet. I've just been researching it as I also plan to do it next September if I get the CSC scholarship. Perhaps someone who actually has done it can weigh in.