For me is a little bit difficult write about places to visit and things to do in my city, because honestly, I think Santiago is pretty boring. Here in Santiago, we don´t have beautiful places to visit like the south of Chile or interesting things to do for a foreigner, because all the things to do are common, there are no originals things to do.
I think the places a foreign should visit are probably Cerro San Cristobal, Cerro Santa Lucía, The National Library, Patronato and Quinta Normal Park.
Cerro San Cristobal is the biggest hill in Santiago, There are a lot of things to do in the east side of the hill (in the west side is the Metropolitan Zoo, but is a pretty ugly and depressing Zoo, so I don´t recommend you to visit it). In the weekends people go to Cerro San Cristobal for many reasons, some sporty people go up the hill running or riding a bike, and other people just walk. There are Tai Chi groups and a Medieval Wrestling school. There is a Virgin Mary statue in the top of the hill and there is a big mass every Sunday. Cerro San Cristobal is a beautiful place to visit and I recommend you to visit it on Saturday or Sunday, wear comfortable clothes and be prepared to walk.
Cerro Santa Lucía is another hill to visit in Santiago, is smallest than San Cristobal but is very beautiful. There are no many things to do there, just walk around and spend a quiet afternoon. In front of the Cerro Santa Lucía is the National Library, it is a beautiful place to visit too, because is a really old building, of colonial times and you can find a lot of books and news papers.
Patronato is a very different place, is a commercial neighborhood with a lot of different stores, mostly clothes stores. The neighborhood is very old and some time ago it was full of Arabian people, but now there are a lot of Chinese and Korean people too. I recommend you to visit Patronato with some money, because you will find a lot of beautiful (and chips) things to buy, but be prepared to walk all day long.
Finally, the last place I recommend you to visit is Quinta Normal Park. It is a very big park and there is the Museum of Natural History, founded by Claude Gay in 1830, so is very old. The museum is not so interesting, but you can spend a nice afternoon visiting the museum and then walking through the Park, enjoying nature and native trees.