I did not realize that next Friday, March 18 is a holiday in Madrid . I will be there and have plans to go to Segovia for the day. Does anyone know if the buses will be running? Are most restaurants closed?

Friday is 19, not 18. And it's San José, not San Diego.

Apart of that, buses will be running, but with holidays' schedule, restaurants in Segovia will be open and probably crowded. Restaurants are the kind of businesses that never close on holidays, but a very few exceptions, even in the smallest villages.