Ray, of the three options described in the forum thread jirit refers you to, I would forget about the metro. Crowded, pickpockets, long ride, changes, etc.
As between the airport express bus Jenny took or the Cercanías train, I think it depends on your timing and where you arrive.
If you arrive at T4 (which is where Iberia, American, and the One World alliance airlines arrive), there is a Cercanías train from the airport at 58 and 28 past every hour (and they are pretty much on time, in my experience). It's a 26 minute train ride to Atocha from T4, no changes. The other real advantage is that there is a RENFE office at the Cercanías stop at the airport, which is rarely crowded, so you can buy your train ticket to Sevilla if you haven´t already done so. If you buy the AVE (fast train) ticket at the airport, you won't get any of the cheap web fares, but if you're over 60 you can get a very nice discount. Also, if you buy an AVE ticket at the airport, they throw in the cost of the Cercanías train to Atocha for free.
I've never taken the airport bus, but since it stops at all terminals, it may be more convenient if you aren't arriving in T4. It has just a few stops between airport and Atocha. Buen camino to you, Laurie