Paella is named after the wide, shallow pan its cooked in, specifically called a paellera outside of Valencia. In Valencia the word paella refers to the rice dish and the pan, and paellera or paellero refers to the part of the house where paella is cooked and the gas burner the paella is cooked on.
Paella is normally eaten for lunch—more specifically on Sundays as part of a family gathering. Aside from tourist-focused restaurants, it's rare to find traditional Valencia restaurants that offer rice dishes at night. If you want to eat paella à la carte in Valencia, make sure to bring a companion: Any good restaurant that makes freshly prepared paella requires a minimum of two people. Some restaurants offer a "rice of the day" that may be paella or another rice dish that doesn't require two people to order it. Read more about ordering paella at About.com.
Want to make paella at home? These instructions by Valencia native José F. Martínez are the most in-depth I've found in English. He's not a professional chef, but being an associate professor for electrical and computer engineering at Cornell may have something to do with his fastidiousness.
All i pebre
Atún con tomate (tuna with tomato)
Horchatería El Siglo: Plaza de Santa Catalina, 11, 46001 Valencia, Spain (map); 34 96 391-8466
Head to Pastelería Santa Catalina for a Coca Cristina after getting horchata from the neighboring Horchatería El Siglo or Horchatería Santa Catalina.
Pastelería Santa Catalina: Plaza Santa Catalina, 7, 46001 Valencia, Spain (map); +34 963 92 28 17