Best free things to do in Santiago?

Valerie, nice mountains close to the city (Farellones, 1 hour each way, cheap), Cajón del Maipo (By bus, cheap), and some nice hills to walk. Museums: Museo Chileno de Arte Precolombino (a must), Cafeteria y Tienda Mseo de la Moda (according to your taste), Vega Central (food market), walk besides mapocho river, República and Yungay, (nice neighbourhoods), Pueblito de Los Dominicos (handicrafts), maybe some vineyard not so far. Valparaiso, a port city is very unique. Enjoy!

Cerro Santa Lucia is a gorgeous park with a mini climb to get to the top. Great photo ops and a nice place for a weekend.

Cerro San Cristobal has some amazing views of Santiago and the Andes on a clear day. I believe there's a shuttle bus but you can also walk (pretty steep, more like a hike) or bike up. It's definitely a half to full day's worth of activity, depending on how long you want to spend up there. 

Mercado Central de Santiago is a giant farmer's market with lots of fresh food and plenty of restaurants with delicious eats! 

Pablo Neruda the famous poet has multiple houses throughout Chile, but the Santiago one is a beautiful museum with really interesting architecture. It's not free but I don't believe entrance was too expensive.

Bellavista is a great neighborhood to walk around. Lots of artwork and cute little shops with artisan crafts. 

Sculpture Park lots of interesting sculptures to see and a good place for people watching as well!

If you want outdoor activies, you can go to Cerro Manquehue or San Cristóbal Hill. The first one is hiking, you climb 500 meters from the start and it has an awesome view to all the city, the second is a small hill, where you can go by bike (30 minutes to one hour) or walking (would be like 2 hours), there is a road. I usually do both, Manquehue hiking and San Cristóbal biking. There are many other hills around Santiago, but these ones are the closest.

If like the outdoors San Cristóbal is Def an Option. If you like art, museums and stuff there is Lastarria