Discourse style learning might be interesting, but unless someone's got DVDs for this, my Chinese isn't good enough to closely follow most live conversations.
My personal theory is vocabulary is the most important thing. If you don't know words, perfect grammar, perfect pronunciation, and perfect (fill in the blank for assorted nuances) don't help. "I want water." grammatically is the same in Chinese, but I believe speakers of either language could get the hint that an aqueous refreshment is desired even if someone said "Want water I.", Water I want.", "I water want.", "Want I water." The listener might even overlook the possible horror movie implications of "Water want I."

Naturally, putting the words in an acceptable order helps, especially for more complex sentences. This will also help native speakers have a better chance of figuring out what word is being said if one word is seriously mispronounced.
I've got a limited basic vocabulary and can make sounds that are at least approximating what I'm trying to say. DuoLingo is very nice about explaining some of the structural differences, like time, then place, then action.
When I'm done with Peppa Pig, I'm plotting to move on to something natively Chinese - Bad Bad Wolf and Pleasant Goat. Hopefully I can get that with Chinese and English subtitles. Because it's originally Chinese, there won't be anything that had to be awkwardly adapted from English.