Take Calle Daoíz, the continuation of Calle Marqués del Arco (where the door to the Cathedral is).
Soon you'll come to Plaza de la Merced. Here you'll find Convento de las Carmelitas Descalzas (Barefoot Carmelites), a convent founded by Santa Teresa of Jesus in 1574, where Saint John of the Cross recited his first mass in Segovia (it's the building on the right, just after you arrive to the square).
If you keep walking you'll come to Plaza de la Reina Victoria Eugenia. Here you'll have very good views of the Alcázar, but also the Cathedral (behind you to the left) and the legendary Vera Cruz church (in the distance to the right of the Alcázar), which it is said the Knights Templars built to guard a splinter of the true cross of Christ. It is highly recommended to return to this square at night when everything is illuminated.
From the Cathedral to the Alcázar takes about 5 minutes.