I have been watching the Dothraki wiki which is accessible through the main DOthraki site, and have not seen a lot of stuff I know to be going on there. But the link you provided above has a lot more material. Are there two wikis about essentially the same thing, or is some updating missing somewhere? In any case, I am glad to see that the information on some of the newer is now available in a systematic arrangement.
But for a total Dothraki noob, and a non-linguist, what is the very first (and second) part of the grammar I should begin to learn?
Which main dothraki site would that be? Dothraki.org? If it is then it's the same wiki. The problem was that starting from the first page there wasn't a clear index page so you could find all the information but that's what we have created now with the link above.
One thing to start with is of course to learn various words or at least know where to look them up when you have decided what you want to say etc. As for grammar though, I think the first thing to check out is how to conjugate verbs. By conjugating verbs you not only know how the verb is used based on who the subject is but you can also express actions in the past and the future etc.
After that, or perhaps even before that, I would check out the syntax which includes things like word order which is needed to know how to string a sentence together.
Then there are several ways to go. Adjectives and how to compare them is always useful to know so that you can not only say hot but also hotter and hottest etc. Noun cases is also something that is needed to be grammatically correct but also it gives ways to express possession etc.
I'll soon start writing a tutorial for creating sentences in Dothraki and put it up on the wiki as well.