Thinking about history:
Since this is a history course taught by a historian, you may wish to read an introduction to some of the components of historical thinking. One such introduction is that by Gerald W. Schlabach, "A Sense of History: Some Components."
Since I continually stress the importance of your learning this history from the "primary sources," I strongly recommend you look at the essay "Why Study History Through Primary Sources."
A survey course such as this one covering a huge area and time span is bound to be challenging. Readings contain numerous unfamiliar names and details. Therefore it is important that you keep asking when you read what are the main themes or main points. You do need to have some command of detail, but generally that which is important is frequently repeated. Reading an item once is probably not enough, partly because you tend not to know what is important until you have been through it the first time. I would recommend first skimming, then going back and reading for more detail. If you let your highlighter do the thinking for you on the first time through, the odds are you have few pages that are not all highlight. The sooner you at least skim a required reading, the better, since if you have done it ahead of when the material may also be covered in lecture, you will get more out of the lectures and may be in a position to ask questions that will help when you return to read more closely.
Examples of readings which may create a real challenge for you are some of the primary sources--Babur, the Tarikh-i Rashidi, the Han histories. In them you often find a welter of names, most of which in fact need not be remembered. However, such sources do illuminate important themes, indicate the nature of relations among peoples, religious influences, and so on. For such sources it is important to have an idea (if it is known) of who wrote them and when, and what the periods and geographical areas are on which they focus.
As you read, it can be important to jot down some notes about names, dates, events that seem to be important. One good device is to create your own chronological tables, where you can have a separate column for each of several topics or each of several geographical areas. That way when it comes to reviewing and synthesizing, you can see at a glance the correlation between events in, say, China and Central Asia, or between religious and economic developments. For major figures or events, precise dates are important, but for many purposes a chronology accurate to part of a century may suffice. For understanding historical development, relative chronology is very important--one must know what precedes what and not get the order reversed. Numerous chronological tables have been provided (and linked to specific weeks' assignments), not with the expectation that you will memorize everything on them, but to give you a framework for understanding events relevant to a given lecture or set of readings.
A good essay has a clear structure whose basic components are an introduction, a middle section elaborating an argument, and a conclusion. Most essay assignments ask that you develop an argument; that argument should be clearly defined in your introduction. That is, you pose a question and offer a thesis, which the body of the essay then will defend. A good introduction guides your reader through the evidence which follows and informs him/her as to why each of your specific points is relevant. You should at least draft an introduction as you are developing the rest of the material in your essay, but it is entirely possible that your study and writing will lead you to a somewhat different conclusion from what you had anticipated. The last stage in finishing the essay may then be to re-write the introduction.
Defending your argument means selecting and analyzing specific evidence, not simply repeating with slightly different wording unsupported generalizations. For an argument to be convincing, it is necessary that you take into account all possible sides of an issue. That is, you cannot just ignore significant contradictory evidence but likely will need to address it specifically, if for no other reason than to convince your reader of why it is not persuasive. Evidence should not merely be a mindless catalogue of facts but rather be a selective analysis of ones which are particularly relevant to your case. To decide what evidence to use, you will wish in advance of writing your essay to lay out the full array of possible evidence. Then you choose that which can best be developed. Where you have primary sources for evidence (for example, first-person accounts written at the time of the events--e.g., Clavijo on Tamerlane's Samarkand), you should prefer them to the assertions of secondary authors (the historians who later write about the subject). Quote only selectively, and quote only that which is particularly valuable as evidence. Quoting a historian's summary of something you could easily say in your own words is not "proof" of anything but the fact that you did not take enough time to digest the material and write your own prose. Papers which are a pastiche of quotations are generally not good papers.
That said, it is important that you give credit where credit is due. Quotations must be indicated as such by the use of quotation marks or (if long) by single-spaced indentation, and a specific reference with page number given as to the source. Paraphrases should also be annotated as such (e.g., a note to the effect that "this paragraph follows Morgan, p. 23"), although in general they should be avoided, since paraphrasing usually indicates you have not really absorbed the reading in order to create your own argument. Unique facts should be footnoted. This does not mean you would have to footnote a statement that "the Mongol Empire came into being in the thirteenth century," but it might mean you would need to footnote a statement that in the year 1252 the Mongols extracted 3521 pounds of silver as tribute from the Russian principalities (if, in fact, you found a source which said that).
While good content in your essays is extremely important, so also is clear writing. Clear writing means good organization of paragraphs, so that they have topic sentences and coherent thematic development, and good transitions from one paragraph to the next. Clear writing also means correct English usage. This means paying attention to grammar (e.g., singular subjects require singular verbs) and to word choice, since it is very easy to think you have said one thing, when in fact you have written something different. When you have finished your essay, run a spell check, but remember spell checks will not catch many common errors.