Look at the incredible 2000 year old Roman aqueduct in the center of town. Take a tour of the large and very decorative cathedral also in the center of the town about a block from the viaduct. Just out of town are two other major tourist sights, El Escorial monastary and palace (lots of gold used here) and the Valley of the Fallen where Franco erected a 492 foot cross to honor the victims of the civil war. There is an incredibly large church under the mountain on which the cross stands which again is well worth taking a tour of (preferably guided so that you get the history of the place). Also between Segovia and Madrid is a large 'Alcazar' a castle from Moorish times which you can tour if you wish. We didn't tour the castle, but we heard it was well worth while.
Quick Tips:
Segovia is about an hour out of Madrid and you could probably cover - at least briefly - all of the highlights above in one day.Best Way To Get Around:
If staying in Segovia, no vehicle is necessary for the in-town touring, but you would want to rent a car or take a bus tour to go to the out of town sites mentioned.