The provincial museum is the best single stop in town and an excellent primer on Patagonian natural history. Three floors cover the geology of Santa Cruz, the fossil record (this is dinosaur country), the indigenous Tehuelche people, and the European settler history of the region. The displays of guanaco, rhea, and pumas are not exactly cutting-edge taxidermy but they're informative.
Entry is free or close to it. Allow an hour. The shop sells a handful of useful books on Patagonian wildlife if you're headed onward to the glaciers or Tierra del Fuego.