What kind of supervision do new captains get on their first flight?
After a new captain is done with schoolhouse training (classroom and simulator) they will fly on-line as a captain and will have a training captain in the right seat.  After some number of flights (for me it was a 4-day pairing) you will be joined by a check airman in the jumpseat and have a line check. 

Once you pass the line check if you are a first time captain you are a high minimums captain for your first 100 hours and then a normal captain. During high mins time you are subject to a few restrictions in approaches you can fly. All the flying after the line check is with normal (low time) first officers.