It can work different ways for different people, and not only did Miss Manners not answer any of them, but she also jumped to the conclusion that the beginning of courtship was the beginning of acquaintance. I don't know why she jumped to that conclusion.

Some people really enjoy the gradual hinting (on the giving or receiving end), the sense of things starting to be a bit warmer in here, and others would prefer to say, "I like you a lot, and I'd really like to find out if a [romantic/sexual/both] relationship between us might work. Is that something you're interested in?" Neither is necessarily more proper taken in context. Obviously there are contexts that render both completely inappropriate.