Can you give best ways to get from Madrid airpot to center of city-it there fast service train? Thanks
The best way is always a hired driver, second best - a taxi - door to door. The airport is relatively close to the downtown area.
I walked out the door of the terminal, crossed the road, walked maybe 50-100 feet and caught a bus that took me to Atocha Station. There is also rail service, but I don't remember which terminal it departs from. Your guide book should have that information.
A taxi from the airport to the center of Madrid is a fixed rate of €30, and there is no need to tip. So, if you're tired, or in a group, that's the way to go. (It's what I did, and I was very glad, jet lagged, to get door-to-door service to my hotel).
You can also take a train, bus, or metro. Details are on the airport's website linked above.
As always, "best" is subjective. And, it will depend on your ultimate destination. For instance, if you're staying within an easy walk of Atocha station, the cercanias train to Atocha is easy. If you're not, a train to Atocha will then require a metro or taxi afterward, making it a less appealing method.
I always take the metro. On my last two trips to Madrid my hotel was about 50 yards from the Puerta del Sol stop.