Welcome to Eorzea

The only thing you need to do for a MSQ to appear is to do the last one. I can't think of any sidequests that are required to move on with the story. The MSQ do have a level on them, however, so if you're trying to turn in a level 20 quest, you won't be able to until you are level 20 on one of your classes (in your case, most likely pugilist).

Do you remember the last quest you did for the main story? It may possibly be in another area.