You can change at Chamartín and take line 1 to Gran Via, or change at Nuevos Ministerios and either take line 6 and change at Cuatro Caminos for line 1 to Gran Via, or take line 10 then change again at Tribunal for line 1, or simply get off at Tribunal and walk a bit. If you're going to take the Metro a lot, get a 10-trip ticket for €12. You'll need a separate Metro ticket: €2, or possibly down to €1.50 if your Metro trip is no more than 9 stations long. If you want to go to a precise Metro station, change for the Metro in the city. You can go from the ticket machines at the airport to street level on Puerta del Sol in 30min, not counting the time to wait for the train at the airport the train only runs every 30min. You can change at Nuevos Ministerios for line 3 or 4 and get off at Sol, the next stop: that's about 300 to 500m from anywhere on Gran Via. Line C-1, which starts at the airport, stops at Recoletos, which is about 700m from the eastern end of Gran Via. The downside of Cercanías is that they stop at fewer places.

These trains are also cheaper than the Metro when you're going from or to the airport (€1,75 for a 3-zone Cercanías ticket, vs €5 for a Metro ticket from the airport to the city).

The fastest way to get into town quickly (barring a taxi, which is expensive and whose speed depends on traffic conditions) is the commuter trains, Cercanías ( journey planner, network map).

Train madrid airport to city center