Alameda Park, just west of the Cathedral, is a well-maintained public park that offers gorgeous views of Santiago. Be sure to walk through the park, about as far as you can to the wester edge. There you'll find a stunning vista of the USC -- that's University of Santiago de Compostela, not the _other_ USC -- that's worth enjoying for several minutes, or an hour.
You'll see plenty of locals lounging around, and plenty of students studying. Plus, it's a great place to keep an eye on the weather, as the rain comes from that direction. You'e high enough so you can see them coming, and close enough to restaurants and bars to head to safety!