For regional Zacatecan food, try restaurante La Cuija, near the cathedral. (Though I, personally, found the regional cuisine rather boring.) Another good, not expensive restaurant is La Leyenda (Av. Matamoros 216 Centro), which I highly recommend.
And don't forget to sample the local wines. Zacatecas is the second-largest wine growing region in Mexico (after northern Baja).
If you're interested in the best-known regional specialty, try Gorditas Doña Julia. It's a chain, but the gorditas are wonderful. There's a location not far from the Cathedral.
For the best place to see and be seen, don't miss the Café Nevería Acrópolis, just across the corner from the Cathedral. Breakfast is delicious, and don't miss the art on the walls.
We haven't been in Zacatecas since 2001, but the restaurants we enjoyed (although not rising to an International level) were Cantera, under the Mercado González Ortega (actually a mall); the Cafetería El Acrópolis, that Cristina mentions, for light fare; El Tragadero, on Avenida Júarez, enroute to the Mina El Éden lower (main) entrance; but also menudo at Menudería La Güera, on Plazuela Genaro Codina, near Mercado El Laberinto.
There's a roasted chicken restaurant, El Pastor, on Jardin Independencia, that was good, but we ate there in 1991.
The birria we tried in Mercado Arroyo de La Plata was very good.