Hi, dduroy,
You have several options in addition to taxi.
There is a bus direct from the airport that goes to Av. Americas -- if you arrive in T4, it'll be Bus 200; in the other terminals, it's Bus 204 I believe. I've done that before and it is really hassle-free and not crowded.
Then there's the metro, take it to Mar de Cristal, where you will have to change to Line 4, in the direction "Arguelles," and from there it's 7 more stops to the Av. Americas bus station.
There are also many buses heading north from Madrid that stop at the airport (but only at T4, I think), so you could avoid going into town. The Alsa schedule shows them pretty clearly, I think.