While looking for moon rise/set times, the times are always listed as
Moonrise, then moonset for a given date. Is the time for moonset for that morning, or is it for the following morning (which would make it the next day)?

The times given are for the date given. Thus quite often moon set occurs first on a given date, follwoed by moon rise approximately 12 hours later - this is ttypical for the phases of first quarter moon through full moon to third quarter. At other times it rises in the morning and sets in the evening - this is typical for phases from third quarter through new moon to first quarter phase. Thus if moon set is listed as earlier than moon rise its because it rose the previous day sometime in the afternon or evening and didn't set until mprnong of the next day.