Here are just a few things we could expect and some links showing how they're demonstrably wrong:
1.) We would expect every humans mitochondrial DNA to share a common ancestor around 6000 years ago. And we would expect to see this in the Y chromosome as well. Instead, we see our common ancestors being much older than that. In fact, we have to go back 150,000 years before we all share the same grandmother.
https://www.genome.gov/27555170
2.) We wouldn't expect humans and apes to share any endogenous retrovirus. Instead, we share at least 19.
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2001186/
http://www.evolutionarymodel.com/ervs.htm
3.) We wouldn't expect any Bristlecone Pine trees to predate the biblical flood (2304 B.C.). Instead, we see some of these trees that were supposedly under miles water for 370 days predating the flood by 500 years.
https://en.wikipedia.org/wiki/Methuselah_(tree)
4.) We would expect there to be no written records of the Sumerians, Egyptians, Romans, Grecians, Chinese, etc. before 2304 B.C. In fact, we would expect them to have no cultural history before that time at all. Instead, we see records of their societies forming long before the flood and continuing for thousands of years after the flood.
https://eprints.soas.ac.uk/3139/1/PAGE_31-71.pdf
http://www.omniglot.com/writing/egyptian.htm
http://www.lib.cam.ac.uk/mulu/oracle.html
5.) We wouldn't expect humans to have any atavisms or vestiges. Instead, we're chock full of them.
https://en.wikipedia.org/wiki/Human_vestigiality
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3014134/
http://www.bio.miami.edu/dana/dox/atavistic.html
6.) As birds were made on the fifth day and land animals weren't made until the sixth day we wouldn't expect to find any land animals in the fossil record before birds.
The evidence here is so common and overwhelming I won't bother posting any links. Birds don't show up until the late Cretaceous Period (about 65 million years ago). This would mean that we shouldn't see any land animals (like dinosaurs) before then. But that's not the case at all. We find them far earlier in the Jurassic Period (145 million years ago) and even earlier than that in the Triassic Period (200 million years ago). We also see other land animals (like amphibians and reptiles) living before long before dinosaurs in the Carboniferous Period (300 million years ago).
The idea that birds came before land animals is one of the most obvious blunders of the Genesis account. If the story is meant to be literal - then it's literally wrong. If it's metaphorical - then it gets the metaphor wrong.