Hold the hat with the crown to the outside in an outside grip. Bring it across the body, and cross the arms. Flick the hat over and drop it onto the thumb of the catching hand. Uncross the arms.
The hat is now held in an outside grip in the other hand, which suggests that the tumble would be a good next move.
The plate is held and thrown vertically, and spins in the vertical plane. Jugglers will recognise this as the way that rings are usually thrown.
From the TricksWithHats archive
Hold the plate by it's edge and spin it so that the rim stays in contact with your thumb. After a single spin catch it again. Once you've mastered singles try longer spins. This trick is very versatile and can be used to add lots of flourishes.
This is one of the videos that was filmed for our former site, TricksWithHats, which never went live. So it doesn't have a write-up and is presented here for posterity.