This is a tough one! If you arrive in Madrid after an overnight flight from the U.S., my advice would be to use that first day to get where you'll spend Christmas eve (nochebuena) and Christmas day (Salamanca?). Then go back to Madrid & do day-trips to Toledo, Segovia, El Escorial. You can even Day trip on AVE to Sevilla (I have done that once). Same day return gets you a discount of about 40%.
Everyone wants to go to Granada (because of the song? Or among the Gay population maybe because it was Federico Garcia Lorca's home) Granada would be more difficult to reach ... it's a matter of priorities. You shouldn't skip Madrid itself (Museo del Prado, Palacio Réal, etc.).
P.S.
Salamanca will be much colder than Madrid. In Madrid around that time you can normally expect lows around zero (Celsius, 32 f) and highs around 13-degrees (55 farenheit). Unless you go to the mountains, you don't need any really heavy winter garb.
To comment on DavidB's advice re: New Year's eve & chiming in the New Year eating grapes at the Puerta del Sol - that's nice! But .. be very, very careful with your belongings. The pickpockets (or purses) have a field day while that is happening. You will be their main target!
BTW
Many Restaurants/night clubs all over Spain will play the sounding of the bells from Madrid's Puerta del Sol at midnight over the loudspeakers - some will even provide grapes for their patrons to eat as the bells chime
[ 11-15-2001: Message edited by: Eddie ]