Would our marriage be legal in both countries and could I then bring her to live in the US after she returns to Vietnam with a marriage visa?

Generally, the USCIS will recognize as valid a marriage that is deemed valid in the jurisdiction where the marriage took place. It would be wise, however, to consult with an immigration attorney who could discuss with you and your girlfriend the immigration-related issues associated with the location of your marriage. If, for example, you became married in the U.S. and filed immigration-related applications while both of you are in the U.S., and then later traveled to Mexico for a second wedding ceremony, you may be able to avoid some of the significant delays and obstacles associated with going abroad to become married. On the other hand, depending upon the type of visa your girlfriend has, marriage outside the U.S. may not interfere with her re-entering the U.S. with the visa she currently has - that is the case, for example, with an H1B visa, but that is not the case with many other types of visas. It would be wise to consult with an immigration attorney who, after learning all of the relevant details about your girlfriend and your goals, could advise about immigration eligibilities, options and strategies. Some immigration law firms, including mine, offer legal services on a "flat fee" basis so that a client will know the total expense from the very beginning, and a few immigration law firms, including mine, offer an initial consultation free of charge.

It will be legal if it is legal in Mexico.