There are of course loads of nice places to see (Museum de la Reina Sofia, Atocha Train Station, Parco del Retiro, El Prado Museum, Templo de Debod and so on) and nice things to do in Madrid, but what I mostly love about the city is the everyday life and this is what I am writing about. Seeing the city the local way it is going to be the best experience!
Mahou is the local beer which you will find in all the bars, try it at any neighborhood bar. Gin tonic is very popular also in Spain, they mix it up very nicely (as far as I heard)
Tapas! you can find anywhere you go. my absolute favorites are the "pimientos del padron" (the small peppers in the picture).
There is a very nice Tapas place, which is called Cafe Melo's it is usually super crowded so you should come with some patience if you want to eat there (and they don´t have places to sit outside) .. but absolutely delicious!!! try the zapatillas, the croquetas and the pimentos, you will LOVE them! This is for sure not for people that care a lot about their diet, although I would recommend for a cheat day. You should also try a "menu del dia" lunch. It is offered in several places, the deeper you go in a neighborhood the better it will taste and the more local the experience you will have.
Some touristic tapas places are El Tigre and Casa Pepe .. even if well known it is quite interesting to go once. Btw.. might be quite an impact, Spaniards tend to trough napkins and food leftovers on the floor, so if you go in a bar and have the feeling it is dirty, you should not worry it is just something very typical thing they do...Spaniards actually get worries when they enter a place and there is not even one napkin on the floor ;)
Gran Via is the place to be for all main brands. Once again if you are an Inditex fan, make sure you come with extra luggage space as the prices are around 10% lower than anywhere else (except Portugal where they apply same rates) and the collections far cooler. On Calle Fuencaral you can find a lot of small boutiques, here you have the chance to finally buy something that you have far less chances to go somewhere and end up dressed with the same dress as somebody else.
Madrid is the home of the very famous Real Madrid and of Atletico de Madrid. If you have the time to watch a game go for Atleti (unless you are a Real fan...) ... as the supporters are some of the best in Europe and the atmosphere is very special. (I've been on Vicente Calderon at a Champions League game Atleti - Barca).
There are a lot of places you can go and party in Madrid, and as we know that Madrid never sleeps the after parties are also quite popular. An interesting place to check out is Maloney's. With oldies type of music, with the ceiling covered with flags and the walls with left messages by people who passed by...the rest of the story you should discover on your own ;)
learn to cook Spanish food. There is a cooking school offering a couple of cooking classes in English. It is called Cooking Point. You can see the result of the Paella class in the picture on the right. The owner is very friendly and flexible.
There are also loads of day trips that can be done from Madrid by bus (cheaper) or by train (faster). Examples like Segovia, Toledo, Avila which are about one hour away.
You will love Madrid no matter what type of person you are!