I would suggest you explore the Salamanca neighbourhood. Starting from Puerta de Alcala, all the way up to Ventas (Plaza de Toros) and then the side streets of Salamanca e.g Velazquez, Serrano and the smaller ones. Plenty of shops and bars to hang out.
Marques de Vadillo area is mostly locals. You may want to be extra cautious if hanging out there.
Templo de Debod is nice too.
You could also do a day trip to El Escorial, for example, in case you have not been there already. I am from Madrid so feel free to ask further questions.