The first contradiction listed in the SAB is "The two contradictory creation stories." No surprise; this is usually the first point any list like this will bring up, because it's the first thing evident in a casual reading. In verse 1:1, God starts creating the world, in verse 2:1, He finishes, and after a breather, He seems to start over again in verse 2:4. What the heck?
First of all, it's quite right to notice this. A person who reads the book of Genesis and fails to notice two separate accounts of creation wasn't reading closely enough. It doesn't take a genius; I think I first noticed this when I was about 9 years old. Even at that age I noticed a lot of the repetitiveness between chapters 1 & 2. Didn't God already do all this?
On the other hand, if you think there's a real problem with this, I don't think you're reading as closely as you could. Yes, there are two stories, but I don't think they contradict, and having two stories serves a very specific purpose for the original writer/editor of the Bible.
The book of Genesis is odd for various reasons, and this particular oddity of the book shows up early: it's not a single story, but a collection of stories put together. Among skeptics of a more scholarly bent, there is a prevailing idea that the Torah (first 5 books of the Bible) was not written my Moses as is claimed by Bible believers, but rather "redacted" or edited together from several sources hundreds of years later. I'm not sure what that has to do with the validity of the content (and maybe they don't intend to impugn its validity anyway), but the concept doesn't bother me much, especially in the case of the book of Genesis. I seem to recall as a child (and as I was raised Jewish, it may be a belief among some Jewish sects) being told that the book of Genesis was not so much written by Moses but edited by him and committed to paper from oral history. This makes sense, since Moses wasn't there when all the events took place, nor were any of Moses' contemporaries.
So, the phrase "These are the generations" that appears in 2:4 is a key to the structure of the book as a whole. Every time you see those words, the indication is, "And now, another story." most of the time, these stories overlap time periods they represent:
Gen 2:4 These are the generations of the heavens and of the earth
Gen 6:9 These are the generations of Noah (cf. Gen 10:1)
Gen 11:10 These are the generations of Shem
Gen 11:27 Now these are the generations of Terah
Gen 25:12 Now these are the generations of Ishmael
Gen 25:19 And these are the generations of Isaac
Gen 36:1 Now these are the generations of Esau (cf. Gen 36:9)
Gen 37:2 These are the generations of Jacob
So, including chapter 1, there are nine main stories. The problem that one encounters with these stories is trying to make sense of them as a sequential whole when that key phrase is telling you, "Now I'm going to go back and tell you this other story." If you doubt that's what it means, sit down and read Genesis 25 and tell me that it makes any sense to read it as a single, continuous story. (V.1, Abraham gets married, has some children. V.5, Abraham gives everything he has to Isaac, a child from a previous marriage. Vv.8-9, Abraham dies, and "his sons Isaac and Ishmael buried him". V.12 Abraham sires Ishmael. V.18 Ishmael dies after having many children and grandchildren. V.19 Abraham sires Isaac. And so on... remember as you reconcile all of this that the Bible states plainly that Ishmael is only 13 years older than Isaac.)
If you want a more contemporary example of non-linear storytelling, check out the movie "Pulp Fiction". (Spoilers ahead: for those who haven't seen the movie but intend to, skip this paragraph.) One of the main characters in the movie is Vincent Vega. In the course of the movie, we see these things happen to Vincent, in order: 1) He enters apartment A, and we do not see him leave it. 2) He takes his boss' wife to dinner, same day, after dark. 3) He is shot to death in the early afternoon of day two. 4) He leaves apartment A, and it's still day one somehow. 5) He and his partner go get breakfast, same day, midmorning. Which do we conclude about Director and screenwriter Quentin Tarantino? A) He doesn't understand concepts like the linear flow of time, the order of meals in the day, and the finality of death. OR B) He's decided the story would be told better in three parts, each of which overlaps in time. I vote for B.
The story that we see in vv. 1:1-2:3 is concerned with the physical origins of the world. The following story, vv. 2:4-6:8 (roughly) is concerned with the spiritual origins of mankind. The time scale is different, the stories overlap, and the details are focused on in another manner. I'm going to focus on only the supposed differences in the order of events, and leave problems with the details in themselves for later.
The easiest one to refute is the assertion that Genesis 1:27 says both sexes of human being were created simultaneously. It doesn't. It says God created humans male and female, but the word "simultaneously" doesn't appear, nor am I aware of anything in the Hebrew grammar that points definitively to this interpretation of events. Nonetheless, on a genetic level, He did anyway, since in creating a man first, one might be fair to assume the presence of both an X and Y chromosome in the first human being. But that's another matter.
A little more involved is the problem of what order mankind, plants, and beasts were made (plants are for some reason omitted from the list on the page linked above, probably an oversight). I see mankind as being created last, definitively. Why does chapter two seem to contradict? In the case of plants, vv.2:5-9 do not say that God created them at this time, but rather that they were lying dormant in the ground, waiting for water. Whether this was upon all the earth, or perhaps only in the area of the garden He proceeded to plant is not clear.
Verse 19, where God seems to create animals all over again is trickier, but involves a translation subtlety. In the NIV, this passage is translated, "Now the LORD God had formed out of the ground all the beasts of the field and all the birds of the air." The question becomes whether God is creating animals all over again, if the stories contradict, or if we've translated in a way that makes it sound wrong. I believe that God had already done the process of creation at this time, and He's only taking a moment to bring the animals to Adam to let him give them names. Sure, it's not 100% clear, but that's not the same as outright contradiction and as a believer I'm certainly willing to give the Bible the benefit of the doubt.