Pack lightly. The town of Zermatt is car-free, so you might have to walk a little with your luggage to get to your accommodation.
The destination Zermatt-Matterhorn has been bestowed with the "Families Are Welcome" Award since 2013.
Zermatt and Gornergrat can be enjoyed year-round. For skiing, you want to visit in the winter or early spring. The late spring (once the snow has melted), summer and fall are terrific times of the year for hiking.