Many restaurants and bars are located outside on the plaza at the end of Calle de los Tristes and have a view of the Alhambra. If you want to try the local wine ask for "un costa", the quality is extremely variable and it is more like sherry (but not fortified) than table wine. Otherwise ask for "vino tinto" (read wine), or "vino blanco" (white wine) de la case (house wine) it is usually good and inexpensive.
Campo de Principe, located in the Realejo neighborhood, has a row of bars/restaurants with outdoor seating. They all have great tapas,so you can't go wrong.
Plaza Nueva has several restaurants and kebob shops. There is often live music.
The Calle Elvira area has many great tapas bars where you can also order meals. One of the best tapas bars is Bella y Bestia. They have huge selection of tapas and many Spanish young people go there.