In verse 1, God repeats the "command to man" to be fruitful and multiply. But something is different this time. Instead of ruling over the animals, it says that the animals shall fear man and that man would now eat animals, but not eat their life, that is, their blood. So this is another effect of the curse: it broke the relationships between people and animals. Now instead of harmony, there is acrimony. I've heard some people suggest that the reason God does this is that the flood destroyed most plant life, and therefore Noah and his descendants had to eat meat to survive. I'm not sure if this is true, given a few factors: 1) the dove found an olive leaf, so obviously not all plants were dead and perhaps some fruit could still be found, 2) if there was not enough plant life to support man, then the plant-eating animals would probably die too, 3) if there were only 2 of each animal, then obviously man could not be eating those animals without wiping out entire species wholesale. The exception to #3 is that there were 14 of each clean animal. Side node: calling them "clean animals" is an anachronism, because we haven't yet been told what is a clean or unclean animal. I'm not sure how to explain this other than that the author is trying to emphasize the righteousness of Noah by suggesting that he did not eat or sacrifice unclean animals, which would be a great offense to the legally-minded priests and Levites. Ignoring that point, even with 14 of the clean animals, and perhaps 20 (?) species of clean animals, I'm not sure if that would be a sustainable food source for the soon-to-be rapidly growing human population. So it doesn't seem impossible, just unlikely, and I still think that this is an intentionally drawn contrast to the command given to Adam in chapter 1, which is much more important.
(Another side note: on the subject of anachronisms, some people suggest that the use of the name, the LORD (in Hebrew, YHWH, commonly transliterated into Yahweh or Jehovah) is also an anachronism, since the name is not "given" until Exodus. There are many possible responses to this, but just note the point.)
Verse 6 and 7 is another poetic passage that emphasizes the importance of blood as life, repeats the statement that man is made in God's image, and repeats the command to multiply and fill the earth. This is more of the repetition that is used to support the JEDP theory (or alternatively, the oral traditions that underly the bible).
In verses 9 and onward, we see another covenant. I didn't mention this before, but covenants play a very, very big role in the literary structure and theology of the bible: literary structure because they are used to logically segment different portions of the bible and theology because the covenants form the structure around which the bible is narrated.
To be brief, these covenants are essentially legal agreements between God and the people who he speaks to: the last one was with Adam/Eve (the Adamic covenant: God granted them dominion over the earth), this one is with Noah (the Noahic covenant), guaranteeing there will be no further floods to destroy the earth. We will see several more covenants later in the bible. When I say "legal agreement", what I mean is that scholars have found structural similarities between the covenants in the bible and Hittite suzerainty treaty forms from the 14th and 13th centuries BCE. Or, according to the JEDP, they generally draw analogies to 6th or 7th century Assyrian legal forms. Either way, it is clear that the Pentateuch was regarded by the author as much a legal work as theological: more accurately, the author is framing the theological discourse in a legal manner. A proper discussion of this is outside of my scope, and we haven't gotten to the most important covenant yet anyway.
Next, we see a rainbow mentioned in the covenant. In fact, the rainbow is the "sign of the covenant" as it says. Note the symbolism: the word rainbow (or the literal Hebrew is "bow", but in context it refers to a rainbow) will appear another three times in the Bible. One of the three times it is used to describe an angel and the other two times it is used to describe God (i.e. there is a rainbow surrounding him). In all of these cases, the rainbow is a symbolic reference to this covenant, which is essentially a covenant of mercy. I say mercy because this is a one-sided covenant (God making an agreement with man without requiring anything from man) and it is a promise not to bring a world-destroying flood upon the earth, no matter what happens. So it is God promising to relent from disaster, which is mercy. Therefore when rainbows are mentioned in the future about God, they are essentially referring to his mercy attribute.
The other thing that's cool about the rainbow is that it is created by the vaporous clouds most closely associated with rain itself. The way I interpret this is that God's wrath is building, it starts raining and it's like God is thinking to himself, "I am going to flood the earth again!" So it keeps raining, but then the rain causes a rainbow to appear, and suddenly God is reminded of his promise, so he relents. It's like this symbol of restraint is built into the tool of wrath itself, and so not only does it remind God of his promise, but it reminds men on the earth that whenever it's raining hard, they can look for the rainbow which is the symbol of mercy and they can know that God will relent. So that's why I really like this passage.
Next is the infamous "Noah getting drunk" story. A lot of people teach various principles from this, but let it suffice to say that this is an instance of a pretty severe indiscretion on Noah's part. I don't pretend to have an explanation why he does it, and the bible would not support any such explanation.
What I will say is that the two different responses of his children (telling others or covering their father) show the strong cultural importance of respecting your elders that existed at the time (and doesn't exist nearly as much in America). It would be pretty much abhorrent in their society and culture to act disrespectfully towards one's father, and as we will see later in the bible, the Law of Moses establishes a death penalty for being unruly towards your parents. Yeah, death.
The other important thing to know about this story is that up through much of the 19th century and possibly into the early 20th century, many people considered Ham to be the forefather of black Africans, and they used the "curse of Ham" as a justification of slavery. In hindsight it is easy to disregard this theory, and scorn those who would believe it, but I really think above all else what this shows is the human propensity towards finding biblical evidence to support what they already believe or want to believe, rather than allowing a study of the bible to form and shape their thoughts. For instance, using this passage to support slavery but then ignoring other parts like the book of Philemon, which clearly speaks out about slavery, is a sure sign of an intellectual bias. And yet such biases can be very pervasive and subtle, so rather than scorn those who used the bible to support slavery, I take this as a lesson to 1) the importance of having solid exegesis, and 2) the importance of always questioning our own beliefs to ensure that we, too, do not slip into the same groupthink, biases and subtle preconceptions about the issues of our time. The various churches that supported slavery are used as an example today by those who are not Christian as a major criticism of religion, and in a limited sense they are quite right. To repeat these mistakes of the past is something that every single Christian has to strive to avoid.
Lastly, Noah dies and we move on to the next epoch of the bible.