Pretty easy and doesn't matter if you booked airfare through Disney. Delta is a participating airline for Onboard Airline Checkin, so if your flight is after 1130am, you can put your checked bags out the night before and they will be checked all the way to your final destination. You don't claim them at customs or check them in at the airport, Disney does it.
They run buses continuously to the airport after the ship is cleared. You go through customs, go to the correct bus line (they have tons of people to help) and they take you to the airport. Even if you have to carry your own bags, it is exceptionally easy and stress free. Disney has a great system. Guest services can even print your boarding passes for you. They explain it pretty well on the ship and there are several threads on this message board covering this topic.