The first thing that always comes to mind with Veracruz is the scene on the zócalo, which on a weekend can go on until the wee hours of the morning. The evening shows presented by the municipal government can be quite good. I've seen some great dance troupes perform jarocho.
Rincón de la Trova (no phone) is a small bar downtown with a real local feel, where on weekends you can hear live groups play son montuno, an old-style Latin tropical music. It's at Plazuela de la Campana, which is simply the widening of the pedestrian-only Callejón Lagunilla between Arteaga and Serdán. Live music is Thursday through Saturday. Friday and Saturday, there's a 35-peso cover.
There are several dance clubs to choose from. Most play Latin music and are in or near the hotel zone. One with good acts is La Casona de la Condesa (tel. 229/933-5451), at Bulevar Avila Camacho 2014.