One of the best things about living in Santiago, is that you're only a couple hours from either the beach or the mountains. From the city there are lots of little hills and mountains in walking distance or a short bus ride away that we like to escape to on the weekends. One of the best areas to go hiking from Santiago is Cajon del Maipo.
From the end of Line 4 on the metro, you can catch a bus that follows a river up into the mountains, surrounded by beautiful views of the Chilean countryside as you draw nearer to the Andes. Another great way to go is to rent a car with some friends, like we did about a month ago. We drove way up into the Andes near Baños Colina, a natural hot spring. We did a short hike up into an alpine valley, trekking across snow fields and surrounded by the kind of view you would expect to see in the Swiss Alps.
The trails there were not for novices or people without good hiking equipment, as the temperatures changed drastically throughout the day and the trails were frequently icy or muddy. Even driving as far as the hot springs isn't a good idea without four wheel drive, thanks to the narrow roads filled with mud and potholes from snow run-off.