My 2 cents...just confirming what others have said and adding a couple more places
Laredo and Garcia de la Navarra are a must in tapas + wine category. Book at least with a week in advance for Laredo; 5-6 days for García de la Navarra. Other options would be Cuenllas (great wine shop too, just beside the bar) or Asturianos
If you choose to go to Garcia de la Navarra, you should head up afterwards to the 6th floor of Madrid's townhall located just by the restaurant. Beautiful roof-top bar and fantastic view of Cibeles, Gran Vía and Castellana
Near Prado a very good, inexpensive place for fried fish Andalucian style and a bottle of fino or manzanilla is La Caleta. You will be transported to Cadiz, in the South of Spain, where the owners are from. Try the Tortillitas de Camarones (Shrimp tortilla), puntillitas (fried small calamari) and boquerones (fried anchovies). I would go to the one in calle Santa Isabel
If you are (and you will) be close to Puerta del Sol a not to miss place for lunch is StreetXO, in Callao's El Corte Inglés department store (9th Floor). It is the 3 Michelin star David Muñoz's second restaurant, with a killer mix of Spanish tapas with oriental fusion. Only walk ins. Good view, by the way
For wonderful sushi and a place to dine, Kabuki is a great choice. They have two venues, both with 1*. The Wellington restaurant is where the owner cooks, although is a tad more expensive. Amazing wine list. Prices are steep but produce is worth it
For dinner, another great option is Sacha, also mentioned above. Nice terraze in the summertime. Confirmed that the wine list is on the modern side and not so appealing, but you can sure find something from Rioja or Ribera. Be sure to reserve 2-3 weeks in advance. Other recommendations for great "traditional" food are, in this order: Viridiana, La Buena Vida, Arce and La Tasquita de Enfrente
Sudestada and Triciclo for fantastic asian fusion; Punto MX is a great mexican place
Enjoy Madrid!