The best eating in Santiago is to be had along the pleasant Rúa de Raiña (off the right side of the Cathedral).
Start with Trafalgar (on the corner of Rúa de Raiña). If you order a draft beer (1.50€), they'll let you try one of their famous Tigres Rabiosos (mussels in hot sauce) for free. If you want a full order, it costs 4.80€.
Then move on to Viñoteca Ventosela, where, if you order a glass of wine, they serve you a delicious piece of "empanada gallega" (sardine/tuna pie). The most famous wines in Galicia are Ribeiro and Albariño (both whites). Reds from the region aren't especially good, though you can always find excellent Riojas and Riberas del Duero in restaurants. Some brands of note are Viña Costeira (Ribeiro) and Martin Codac (Albariño).
To finish up, try the most authentic of all: El gato negro (it's a good idea to arrive before 21:00 to avoid problems getting a table). This classic bar serves high-quality food at very good prices. The steamed "berberechos" (cockles, 6€), "jamon asado" (baked ham, 10€) and "mejillones en escabeche" (mussels in pickling brine, 5€) are phenomenal. Accompany it with a jug of Ribeiro (7€). Dinner will run you around 15€ per person.
If you're still hungry or couldn't get a table at El Gato Negro, you can always go to María Castaña (just to the right of Gato Negro) and enjoy their tasty Chipirones María Castaña (small cuttlefish, 7.40€).
If the weather is nice, in Plaza Fonseca (on Rúa Raiña, near the Cathedral) there are very nice places for having an after-dinner drink.
