If memory serves me, the Museo Naval and the Prado are on the same avenue, both well worth a visit as is the Palacio Real. It's a big city, you could spend a week there and still not scratch the surface much.
I have been there 3 times, a week on each occasion, and passed through on another occasion via Barajas on a tour of Extremadura (would you believe another week :-) ), now you have me wanting to go back so maybe time for the AVE to come to the rescue and whisk us up there for a long weekend!
Fond memories of places such as the Plaza Mayor (especially at night), Restaurante Botín and Chocolatería San Ginés (both just off the Plaza Mayor), plus countless other places whose names I have forgotten but could still find my way back to, Sol, Gran Vía, Estación de Atocha, the list is endless.
And yes, the metro is extremely good and great value for getting around the city.
Go for it,
Michael.