Hey James!
Soccer will be in season when you're there, despite the chilly weather. The most famous team (by a country mile) is Colo-Colo. They play their home games in Macul, which is south east of Santiago.
The home stadium of the national team, Estadio Nacional Julio Martínez Prádanos, is occupied by Club Universidad de Chile (also in Santiago). This is a more impressive stadium and there's a lot of history, some joyous and other grim (you'll notice this is a trend). It hosted World Cup games in the 1960s and was also notoriously used as a detention facility during the Pinochet regime.
Most of the top-flight soccer teams are based in the Santiago area, though the Santiago Wanderers play in Valparaiso. The toughest thing to navigate is purchasing tickets (I ran into this problem). Even with a fluent Spanish speaker helping me out, I struggled to secure tickets. If you're coming from a country outside North America, this may be less surprising to you as it's not uncommon for clubs to restrict access to members/supporters/etc. until the last minute, if at all.
I was not successful in securing tickets, but I also could only realistically go to games in Valpo and, we had other stuff going on. Other sports are certainly popular in Chile, particularly basketball but it didn't appear there was much of a league structure in place. The national sport in Chile is rodeo and there is a big cowboy culture in the country despite it being less agrarian than it used to be. Anyway, if you can find your way to one of those... do it. It's incredible and the vibes are excellent.
Anyway! Hope you have a great trip.