Carla, there is a wonderful travel agency in Madrid which will arrange any type of day or multiple night tour that you like. On-line they have listed Segovia as one of their choices, and they also have a full day tour of El Escorial and Segovia. You could skip the GCT optional tour of El Escorial and put the money toward this full day trip, and it would give you more time in Madrid, too.
Since we drove to Segovia from Madrid, I can't tell you how to get there by public transportation. But these people can! You just tell them what you want, and they will make it happen. They are extremely flexible, very reliable, and their priority is pleasing their clients. They can arrange train tickets, rental cars, hotels, etc. We have used several of their private guides for morning or afternoon tours, and each one was excellent.
Our time in Spain has always been on our own, with travel mostly by rental car, sometimes by train or air. They have been extremely helpful to us, and we have referred them to friends who have been happy as well. Take a look at www.madridandbeyond.com.
The aquaduct ruins at Segovia are wonderful, there are several good restaurants, the cathedral, a castle with fabulous views, etc. And if you decide to go for a night, do stay in the old medieval part of the city up on the hill. Have fun!