This is easily one of my favorite barrios in Madrid, and is the first place I stayed while couchsurfing. It has a high immigrant population, and a hippy artsy feel. This is the neighborhood to go out for Indian or Senegalese food. There are many places and most all of them are delicious, but here are some of my personal favorites.
Calcutta, C/ Lavapiés, 48, 28012
Shapla 2, C/ Lavapiés, 42, 28012
Bar Colores, C/ Mesón de Paredes, 43 28012
Baobab, C/ de los Cabestreros, 1, 28012
Cafes: (these both happen to be adorable bookstores and have wifi as well; a very common kind of cafe in Madrid)
La Libre, C/ Argumosa, 39, 28012
La Fugitiva, C/ Santa Isabel, 7, 28012
I recommend all of Calle Argumosa for tapas and drinks in Lavapiés. There are many cute places to go for a tinto de verano or vermut de grifo.
Bar Automatico, C/ Argumosa, 17, 28012
Casa de Asutias (ask for una botella de sidra to share!) C/ Argumosa, 17, 28012
Mercado San Fernando, C/ Embajadores, 41, 28012 :
Visiting this market is a true Spanish experience. It is a building full of shops and stands selling traditional artisan goods such as olives, pimentón dulce, jamón ibérico,and orujo. There is a good quality and affordable wine shop that will refill your wine bottles when you bring them back. There are also food stands. I highly recommend the Greek food stand near the back of the mercado. On Sundays there are usually people dancing in the little plaza within the mercado; one week I saw polka and another some sort of Americana folk dance thing. On Sundays there are lots of locals having tapas, drinking beers, and having a nice time.
Centro El Horno, Calle de la Esgrima, 11, 28012
I was a dedicated student at El Horno, a wonderful dance center near the metro Tirso de Molina, where I religiously took zumba class each week from Richard Quintana, and had just discovered the amazing Bollywood class taught by Ender Bonilla when I moved out of the city. I recommend getting an abono membership. There are many class options in this apartment complex converted into a dance center. These range from ballet to capoeira to pole dance to yoga to spin classes, and the basement is a complete gym.