From Barajas airport one can take a bus to Madrid, that stops right infront of the International arrivals lounge. This bus is fairly frequent (15-30 minutes periodicity) and its last stop is at Plaza de Colon, close to the center of town. If you continue to the University, you should go to that last stop, and locate the entrance to the underground train station called Recoletos, also at Plaza de Colon. You can pick a train that takes you from that station to the University.
You can take a taxi from the airport to town, by approximately 2000 ptas (20 dollars). If you are continuing to the Universidad Autonoma, perhaps the best choice is to ask the driver to take you to Chamartin train station, and take a train from there to the University. Obviously you might as well pick a taxi from the airport to the University, which would cost you around 4000 ptas.
There are buses from Madrid to the University, which are fairly frequent at typical high hours (Here is the time table. ) They leave from Plaza de Castilla. You might reach Plaza de Castilla by metro (there is a stop there). Then, exit the metro station and locate the terminal of buses going to the University. Here is a map of the metro
Probably the best way. There is a train station at the University. The line is the one from Madrid to Tres Cantos. Trains to Tres Cantos ( a small town north of Madrid) start at Madrid South Train Station called Atocha. The trains go underground along the Castellana street. They stop at Recoletos (Plaza de Colon), Nuevos Ministerios (close to the Consejo Superior de Investigaciones Cientificas) and Chamartin station ( The main North Train Station). After Chamartin, the second stop on the way to TRES CANTOS is Cantoblanco-Universidad, where you should exit. Please consult the present Madrid- Tres Cantos and Tres Cantos -Madrid train time-table. Click here for a cleaner version of the time-table.
For your help, here there are a couple of small maps of the campus: map1 and map2 ; and a photo of the University Campus.