Hi, Sharon,
I arrived in Madrid last May and took the AVE train to Sevilla on my day of arrival just as you are planning to do. I had to run a few errands in Madrid beforehand (to the Barrabes store to buy new walking sticks and to movistar to get a phone), so I got a later train in the afternoon.
I think the main thing you need to decide is whether to buy a train ticket ahead of time online, which can mean a lot of savings (33 as opposed to 80+ euros) but gives you a non-refundable, non-changeable ticket. Since I knew I would be spending several hours in Madrid, I booked a train leaving around 3-4 pm and found I had time to spare. So I used the time to buy food for my first day's walk and to say hi on the internet. But with the walk-up option, you of course get much greater flexibility. There are trains almost every hour, I think.
Public transit to Atocha has just improved with the new Airport Express bus. I think it's 2E and goes from the airport non-stop to downtown, with stops at O'Donnell, Cibeles, and the Atocha train station. That would be my first choice. Of course you're hostage to the traffic -- but with good luck you should be there in 40 minutes.
A second option is the metro, which requires 3 lines (two changes) and takes about an hour or maybe a little less). And one final possibility is to combine the metro with the RENFE commuter lines, called Cercanias. I used it to get from Puerta del Sol out to the airport last year and it was much faster and easier. And just a euro or two more expensive.
Buen camino, Laurie