What are some recommendations for things to eat & see in Madrid if I'm there for 5 days
Madrid is one of those cities that rewards the curious eater above almost anyone else, and five days is a genuinely good amount of time to get past the tourist circuit and into the real thing. Start by getting your bearings in La Latina and Lavapiés, which are the neighborhoods where Madrid feels most like itself, old tavernas, vermouth on Sunday mornings, markets that are actually used by locals. The Mercado de San Miguel gets a lot of attention but it's mostly a tourist trap at this point; head instead to the Mercado de la Cebada or, better yet, wander into any of the old-school bodegas around Cava Baja for a glass of something cold and a plate of jamón.
For restaurants, you have to spend at least one night eating very, very well. If you can score a table at DiverXO, that's David Muñoz's three-Michelin-star flagship and one of the most singular dining experiences in Europe right now, it's theatrical, it's polarizing, and it's completely worth the hype if you're open to being surprised. If that's either too booked or too much of a commitment, RavioXO is the move. It's Muñoz's more accessible project and it does not take itself too seriously on the surface, but the kitchen is dead serious, the negronis are excellent and the cooking, especially anything involving seafood, is genuinely exciting. That's your splurge sorted. Round out the rest of your eating with cocido madrileño at least once (Malacatín does a proper one), long lunches with wine, and late-night tapas the way locals actually do it, which means after 10pm, always.
On the cultural side, the Prado is non-negotiable, give it a proper half-day and don't try to see everything, just go deep on Velázquez and Goya. The Reina Sofía is excellent too, and seeing Guernica in person is one of those moments that stays with you. For a day trip, Toledo is close and completely worth it, but if the weather is good and you have the time, the Sierra de Guadarrama is a beautiful escape into the mountains less than an hour from the city.