You can spend some time in Hinterdorf, the oldest section of Zermatt. This is the original cow village, with buildings from the 16th and 17th century, long before Zermatt became famous as the base for starting climbs of the Matterhorn.

Here you'll find a fountain, dedicated to local climbing guide Ulrich Inderbinen. Herr Inderbinen summited the Matterhorn 370 times, the last when he was 90. The fountain and plaque were dedicated to him on the occasion of his 100th birthday.

And you can take a walk to the Gorner Gorge, just about an hour round trip from the center of town. Not a place for the acrophobic, though. Note the section of the trail in the upper third if the picture.