How to get from the Madrid airport to Atocha train station

Have you tried the airport's website? www.aeropuertomadrid-barajas.com/transportation/madrid-airport-bymetro.htm

There's an Airport Express bus. 5 Euros. Runs often. Here's a link: http://madridman.com/blog-madrid/2012/01/madrid-airport-express-shuttle-bus/

We took the Cercanias train ( C-1 line) to/from the airport to/from Madrid's Puerta de Atocha station. Travel time is about 25 minutes & trains travel frequently to/ from the airport.

If you will be taking an AVE train from Atocha, ( you can buy your tickets at the airport's Renfe office) your Cercanias ticket will be free. Bear in mind that the train is not affected by street traffic, like the shuttle bus.
You may read about this form of transportation here: http://www.aeropuertomadrid-barajas.com/transportation/madrid-airport-bytrain.htm

Have a wonderful trip!

As always, it depends on what you mean by "best."

It also depends on why you are going to Atocha. If you're going there to take another train, it makes sense to use Priscilla's recommendation, and get the free Cercanias that comes with the railway ticket.

If you're staying in Madrid within walking distance, the train and the bus both should work well. There are also Metro connections, but these are inconvenient as they require changes.

On the other hand, if you're going there to catch a cab to your Madrid hotel, consider taking a taxi all the way from the airport. It's a fixed fare of 30 euros to the center (a large zone) and no extra tip is required. Particularly for more than 1 person, or if you have larger luggage, or are exhausted, it's money well spent.

The Airport Express is the best way, in my opinion. You get it just outside T4 or T1, and the only places it stops are Plaza de Cibeles and Atocha. It costs 5 euro, and takes about 25 minutes.