I understand in Genesis the progression from Adam and Eve, to the flood, and then to the Tower of Babel. But from where did different human races come? Did the Lord also change our DNA when He confused our languages? Furthermore, how did each race end up in different locations around the world?
The Bible is clear in teaching that the scattering at Babel was entirely and exclusively a matter of confusing human language:
There is no reason to believe the Lord supernaturally changed DNA or any other physical trait during the flood, or at Babel. To learn more about the biblical view of race, please read the following article: Bible’s view on race and interracial marriage
Mutation and selection explain the expression of different physical traits between populations of human beings. The Lord could simply have designed diversity and adaptation into Adam's DNA. We can see an example of this in the variation of dogs that we have been able to breed, even in very recent history. We have, in a very short period of time, turned what would have been an original canine species (from the Ark), like a wolf, into a large variation of dog breeds (or races for dogs). This same process (mutation and selection) is what causes variation in human populations and speciation for all other kinds of animals and plant life. In the case of humans, Babel was simply a tool for dispersing us across the globe, likely in family groups; those with similar genetic traits established since Noah, travelled together. For a more thorough explanation of the genetic consqeuences of the biblical events in Genesis, please read Noah and Genetics. We also recommend our Evolution vs. Creation seminar.
Regarding how people travelled from Babel to locations such as North America, see the following article in the Journal of Creation: The Ice Age as a mechanism for post-Flood dispersal.