Museo Nacional del Prado (https://www.museodelprado.es/en/)
Thyseen-Bornemisza Museum (http://www.museothyssen.org/en/thyssen/home)
Museo Nacional Centro de Arte Reina Sofia (http://www.museoreinasofia.es/en)
Madrid offers a great variety of events that will suit any interest- from expositions to concerts, sports to theater.
For a complete list of daily events: http://www.hoyenmadrid.info/es/home
For an overview of all events: http://www.esmadrid.com/calendario-de-eventos/
Platea (turned a historic movie theater into a dining hall with various food stands offering an amazing variety of food from iberico ham, to spanish omelette, croquetas, cocktails...a must see in Madrid), Retiro Park, Plaza Mayor, Royal Palace, Temple of Debod, Palacio de Cibeles (Palacio de Cibeles Madrid Spain. The cathedral-like landmark was built in 1909 by Antonio Palacios. This impressive building was home to the Postal & Telegraphic Museum until 2007 when the landmark building became the Madrid City Hall) Plaza de Cibeles (This is where all Real Madrid fans reunite after they win- located in front of City Hall), Gran Via (great shopping!), Santiago Bernabeu Stadium (If you are a Real Madrid soccer fan, you have to visit their stadium!), Basilica de San Francisco el Grande, Sabatini Gardens (right next to the Royal Palace), Gate of Europe, Puerta del Sol (all roads in Madrid lead back to this square)
Barrio Salamanca- in this neighborhood you will find everything you need in terms of shopping and design.
El tocador de la novia (vintage wedding dresses), el baul frances (boutique), Enfant Terrible, La nun, Dolores Promesas, El Corte Ingles (huge department Spanish department store- I recommend the one on La Castellana, it is the nicest and largest one), Ursa Lolita, Bimba y Lola, Uterqüe (high end version of Zara), Castañer (famous Spanish sandals), Aristocrazy (jewelry)
Furniture/ Decor: La tienda cuarenta y dos (decorative items), A-cero In (mix of modern unique pieces, curated by famous architects Rafael Llamazares and Joaquín Torres), Pepe Leal (eclectic pieces), Bazar, Teresa Galan, Luis Rodriguez (high end furniture), Banni (italian modern), L.A Studio (eclectic), Distrito (primarily Italian designers), Candelabro (antiques), Skimo (antiques), Berenice (antiques, one of a kind pieces), Slou, Becara (jewelry, furniture and decor...I love everything in the store, this is a must see!), Mis en Demeure (founded in Paris and have various stores worldwide, they have unique, antique looking pieces), Zara Home (perfect for basics!)
El Babero, Casa Lucio, Del Diego (famous for their cocktails!), Vi Cool, Luzi Bombon, Platea (by far one of my favorite places!), Volvoreta (eurostar), Viridiano, Ten con Ten, Otto, Lagrimas Negras, Sergio Arola Gastro, El mentidero (bar), Arte 14, Palacio de Anglona, Bendita Locura, Valor (A MUST! Order Chocolate caliente con churros...you won't regret it), Terraza Cibeles (great view of Plaza de Cibeles)
If you are interested in a different type of dining experience, I definitely recommend going to a Mercado- it is a market that offers fresh produce as well as booths that offer a variety of tapas and other food options.- Mercado de San Ildefonso (the newest one, I recommend getting something from each stand for more variety), Mercado San Anton (the rooftop bar offers some very good cocktails) , Mercado de San Miguel (the oldest city market which offers the most authentic experience).
Nightlife: La Posada de las Animas, Gabana 1800