Assuming you've finished the main path with your first visit to The Institute, this orientation day at the Prydwen is a similar to meeting the new team during the Institutionalized quest. You can talk to these guys in any order. You can see Proctor Ingram above.
This is Proctor Teagan. He'll be your valued trader at the Prydwen.
Then look for Proctor Quinlan. It'll be a pleasure to make your acquaintance.
Finally, this is Knight-Captain Cade. He will conduct a very quick medical check-up.