I just did this one week ago for the first time. I chose to take the train and had a great experience. But, I did my research first.
We took the blue Metro line 1 to Chamartin station. There, we took the AVE train to Segovia for 10,50€. It takes less than 30 minutes and spends a fair amount of that time in three tunnels. When we arrived at Segovia Guiomar station (in the middle of a cow pasture) there were two Segovia city busses waiting for us. We took route 11 for 1,03€ each. Route 11 takes about 20 minutes to reach the aqueduct which is the end of the line. We crossed the street and headed toward the aqueduct (can't miss it). We noted that the route 11 stop for the return to the station is on this opposite side of the street from where we were dropped off. Times are posted and we noted which one would get us back to the train station in time for the return train.
In the shadow of the aqueduct is a sign with the map of the city with sights and several walking routes marked. We noted the blue walking route would lead us from our current location at the aqueduct to Plaza Mayor, the Cathedral, and Alcazar. If you need more info, walk under the aqueduct and you will see an info office on your right. The walk was slightly uphill, but not steep. It took us past lots of cute shops and restaurants. Once at Plaza Mayor, we toured the inside of the Cathedral (3 € each). Then, we continued the walk now downhill toward the Alcazar. The ticket office is to the left. The entrance fee is 5 € or 7 € if you wish to climb the tower (torre). You can add a tour guide to your entrance fee for 1 € more. After our visit, we walked back to Plaza Mayor, but then veered left so that we saw a few more sites and shops. We ended up at the top of the aqueduct with a nice view. We walked down the stairs to the plaza and back to the bus 11 return stop.
Segovia city bus - http://www.urbanosdesegovia.com/index2.htm
Alcazar de Segovia - http://www.alcazardesegovia.com/#