Raised as a SF Bay Area foodie, I have lived in Granada on and off over the past 20 yrs. Here are my favorite places:
Bodegas Castañedas (off of Calle Elvira but NOT la Antigua Castañeda which offers cheap, low quality food despite the pleasant outdoor seating; rather, the indoor one on the other side). Vermouth is a local favorite, the gazpacho is homemade (although not served with a tapa) and beer is most popular. Tapas are hot at midday, but still tasty and varied at night. After 20 yrs and despite the growing tourist crowd, this authentic spot is still my favorite.
Los Diamantes in Plaza Nueva. Almost exclusively fried seafood, but very fresh. Tasty tapas place, although the "raciones" are also worthwhile. Some outdoor seating.
La Tana (off of Calle San Matías). I go there for their broad selection of wine-by-the-glass, atmosphere and interesting twists to traditional tapas. Small, so get there "early" (9 pm) if you hope to grab a stool.
Charming Café 4 Gatos (at the top of Cuesta San Gregorio in the Albaicin) that serves real 100% arabica coffee, not the usual "torrefacto" stuff which is undrinkable. When I want healthy and high quality food, I go here; in addition to the traditional breakfast, they offer muesli, granola and organic whole wheat toast. The homemade pastries are great. Decent tapas and a local Pale Ale for alternative beer lovers! Ideal for Alhambra gazing and people watching despite the limited outdoor seating.
Café Fútbol (Plaza Mariana Pineda) is famous for their "churros & chocolate" and have very pleasant and ample seating in the plaza. When I want churros, I generally go here.
Tourist food/tapas traps to avoid: Plaza Nueva proper & Paseo de los Tristes.