The truth of the matter is that, like any other major travel destination, Madrid's cuisine and nightlife opportunities downtown (i.e. near the tourist attractions) are a bit more expensive than your usual out-of-the-way dive bar. In Madrid's case, the hike in prices also includes a hike in quality and quantity, particularly when heading south of the Puerta del Sol toward Plaza de Santa Ana (the Huertas neighborhood), as well as a collection of live music institutions between Calle de Alcala and the famous Gran Via.
There are some great tabernas in Huertas, serving a mix of locals and tourists the usual Madrid combo of tapas and beer. As far as authentic Spanish food that doesn't cater only to tourists, try Lamucca de Prado. Also, try the rooftop bar at the ME Hotel for drinks and a stellar view; you'll feel like you're riding in style.
Another of Centro's many appeals is its selection of music. Giants like Joy Eslava (Calle del Arenal), Wurlitzer Ballroom (Calle de las Tres Cruces), Sala El Sol (Calle de los Jardines) and Costello Club (Calle Caballero de Gracia) are all big spaces that cater to big name acts, while Mi Madre Era Una Groupie (yup, you read that right: My Mom was a Groupie) is a throwback shack in Huertas worth a gander.
Tapas, music, and a smattering of wine bars, lounges, and more make Centro a great place to hang out without having to go far from the sights and sounds that drew you to Madrid.