Welcome to Madrid, Spain
Madrid is a large capital city with both modern business areas and quaint old streets. I found it to be superior to its rival big city, Barcelona, but there is a lot of disagreement on that comparison. I spent a weekend here on my way to begin a Spanish Language program in Cádiz.
Take a walk to see the sights. It's helpful to have a guide to know what you are looking at. I was lucky enough to have wonderful Couchsurfing hosts as tour guides, but if you are not as fortunate there are plenty of free walking tours to be found. For example, this one, or this other one. Most likely there will be tours advertised at your hostel as well.
Disclaimer: These tours are not exactly "free." The guides work for tips so the idea is to pay what you can.
EL RASTRO MARKET
Browse through this open air flea market with offerings ranging from clothes to jewelry to antiques - Go, wander, and see what you find.
Location: Along Plaza de Cascorro and Ribera de Curtidores
Open every Sunday from 9-3
Eat a huge meal of Tapas - Salmorejo, croquettes, jamon, pulpo, paella... Eat all of it because of course the best thing about travel is the food. This can be done fairly cheap at the right place. The best thing is the meal is family style - I recommend rounding up a few friends and ordering a bunch of dishes to share.