Take the Airport Express bus....it will be outside the exit from the luggage pickup spot. If you do not see a sign, ask one of the airport personnel..Cost is 5 euros or so, cash, paid as you get on the bus. You take the luggage on the bus with you, there is adequate storage for it as you enter the bus. Time of travel is 25 minutes, only one stop at Cibeles as I recall. And you will be deposited at the front door of Atocha. I agree not to purchase your train ticket until you arrive...planes are late sometimes you know and tickets are not refundable. The ticket will have the car number on it. As you board the train, look for the location of the beverage and food car...sandwiches, etc. The trip is 2 1/2 hours. Relax and enjoy the trip. Limited xray of suitcases as you enter the departure area.
If you purchase your tickets at Atocha, when you enter, get a number from the machine, just like in an ice cream shop here in the states. Have written down that you want a one way or two way ticket to Seville...most understand, but it is better to be safe. Escalators and elevators from the departure lounge to the tracks level. Not sure of location of Ticket booth in the airport.
Travel second class ticket, not first class....it is luxury enough!!