We're given so much reading and I have no idea how to get all the details. The web says to understand the big picture first and then go into the small stuff but how do I do that? I'm not even sure if the unit I'm reading will show up on Google. Should I read the pages once and after I've gotten a very very broad understanding of what happened I read it again and pick up on the other things?
Yes, I would suggest doing at least one good solid read through. You may find that certain portions are stressed more - or perhaps your prof gave you some hints as to what they thought was more important, etc, and you will want to devote some additional time to those areas. Depending how much you enjoy reading history, a second readthrough could be a good thing or a bad thing!
Try to be cognizant of any watershed moments that occur in whatever period you're studying. For example, one could consider the Council of Nicea to be a watershed moment in the later Roman Empire, as the adoption of Christianity changed the face of Rome from thence forward. The storming of the bastille might be considered a watershed moment for the French revolution - and indeed, for the early modern era. Perhaps the battle of waterloo is another, etc.
If you're the type to make notes, and you find it helps, you can definitely do that and then use those notes to study from. It's much easier to study from a page of scribbled notes in a pinch than it is to carry around one or multiple textbooks. This type of note taking is ideal if you're looking to memorize a few key names /dates as well.
I wouldn't sweat memorizing dates too much, but, it does look really good if you can plug a few dates in there. I can still spew out that in 800 CE, Charlemagne was crowned Emperor of the Romans by the Pope on Christmas day in Rome. I thought that was important in my medieval history course, and figured, there's a chance I could use this on the final exam in some form.
Don't worry about trying to quote from primary sources during an exam, that's not really feasible and I don't think a prof would expect that level of memorization. But, absolutely be aware of who the primary sources are (who were they, why were they writing, for what purpose, for what audience, potential biases).
Though I don't think it should be a primary consideration during studying, one could devote some time to studying the historiography of the period as well. What were the opinions in the past, how have they changed with respect to the current accepted view, etc.
I hope this helps!