If you're looking for a city in Spain to eat on the cheap and in great variety, then Granada is one of the best places to visit. There are many small plazas around the city known for their outdoor seating and sundry more establishments that will get you free tapas with your drinks; Granada is also one of the rare cities where you can walk a block or two from its more notable downtown attractions and not have to pay tourist prices at most places (although any nice restaurant with a view of the Alhambra will set you back a few extra Euro at least).
The best area to stroll if you're looking to eat, especially if you're at a snacking pace, is Calle Elvira as it runs north from the Plaza Nueva. There is plenty of outdoor seating, cheap tapas, and typical Andalucian grub for all tastes in this area. Bodegas Castañeda, Bodega La Antigualla, and La Bella y bestia II all have reputations for extraordinary tapas, while Restaurante Los Manueles and Restaurante Bar León are authentic Spanish classics. The pedestrian streets maybe half a dozen blocks south near Calle Navas are also home to a nice selection of restaurants along the same lines; although Los Diamantes shines for its seafood tapas and great local feel.
If you head to the Albaycin neighborhood just east of downtown, you'll often get lunch or dinner (remember, Americans, that these meals are typically three hours later than you are used to) with a view over the Alhambra. Restaurants near the Mirador de San Nicolas like La Estrella De San Nicolas and Bar Kiki will be expensive meals that benefit immensely from the view. Other parts of town where you'll find a great concentration of food include the streets around the Plaza del Toros at the northern end of the city and Calle Gonzalo Gallas on the western edge of the university campus, both of which are popular for (you guessed it) tapas joints and Spanish/Mediterranean restaurants.
As you can see, you'll get a chance to try a lot of tapas and get a few genuine, Andalucian meals (fish, Iberian ham, a cold vegetable soup called gazpacho, and everything and anything that could possibly go in the fryer). Combine tasty, cheap (especially compared to Barcelona and Madrid) meals with cultural delights and proximity to adventure and you'll want to make Granada a place to enjoy not once, but repeatedly for many years to come.