If one of my data sets is normally distributed and the other is not, can I do a Mann-Whitney U-test on them, or would they both have to be non-normal?
For the Mann-Whitney $U$-test, the distributions can be arbitrary (any old thing--they do not have to be normal). However, the test does assume that the distributions would be identical under the null (e.g., both chi-squared with the same parameters). If you believe the distributions are not the same and certainly would not be, even if the null hypothesis were true, then it is not clear why you are testing if the distributions differ or what exactly you are trying to test. Tests can still be conducted to scrutinize less common aspects of the distributions (making something up: to see if two different types of distributions, but with the same mean, differ in variance), but it would require some hard thinking to determine what such a difference would mean and how to validly test it.