Hi, Jllpole, welcome to the forum!
What is the proper etiquette? US Anthem first or the other country?
The civilian flag code does not address this question. US military regulations state that the US anthem will be played after other nations' anthems when they are performed together. For example, Navy regulations say:
When a foreign national anthem is prescribed in connection with honors, and it is considered appropriate to perform the National Anthem of the United States therewith, the National Anthem of the United States will be performed last.
However, there is an exception. When the flag is raised in the morning on a Navy ship, and ships of other nations are present, the US anthem is played during the flag raising, and then the anthems of the other nations.
These military regulations don't actually apply to non-military situations, but they are good guidelines. I'd say that you should play the US anthem last, unless playing the anthem is directly related to raising or presenting the US flag. In that case, the US anthem should be played while the US flag is being raised or presented.
I hope this helps!