Segovia is 90 kilometres northwest of Madrid and one of the most accessible day trips from the capital. The fastest option is the high-speed AVE train from Madrid Chamartín, which reaches Segovia–Guiomar station in about 30 minutes. This is the easiest way to visit. From the station, a bus or taxi takes you to the old city in ten minutes — it is some distance from the walled centre.
Regular (non-high-speed) trains from Madrid Chamartín also run to Segovia, taking about two hours via a different line. Buses from Madrid Moncloa bus station take 75-90 minutes and drop you closer to the city centre, making them a decent alternative to the train. By car from Madrid, take the A-6 northwest and then the AP-61 — there is good parking outside the walls.
There is no reason to stay overnight unless you want to see the city in the evening light without the day-trip crowds, which is a perfectly good reason. Segovia has several small hotels in the old city.