At our school, the ceremony involved a series of speeches, and ended with everyone in the class filing across the stage one at a time. They put our white coats on us at one end, and read out our names, undergraduate institutions, and degrees as we walked across. As our reward for not tripping and humiliating ourselves on the way across the stage, we got a book and a photo op on the other side.
Many schools hold their ceremonies after the students have already had a chance to meet. Our school held it
at the end of orientation week (which was plenty of time to be non-awkward), and there are others that hold off for a semester or more before even giving their students white coats. I think it generally depends on when students are going to begin seeing patients - the earlier patient contact starts, the earlier the schools break out the coats.