From Plaza Mayor we strongly recommend walking all the way up Calle Toro, the most popular street in the city and brimming with people and shops (7 minutes). Then, turning right onto Avenida de Mirat and continuing until Plaza España, you'll come to Birdland, a nice alternative establishment with two levels with a Moorish decor and good views of Alamedilla Park (coffee 1.20€; soft drinks 2€).
If you walk along Avenida del Mirat to Puerta de Zamora (5 minutes), you'll come to Iglesia de San Marcos, a beautiful Romanesque church with a circular floor plan. On the way, note the precious palace-library on the right in Plaza de Gabriel y Galán.