The Gornergrat is accessible all year round by the Gornergrat Bahn.
Travel time:
From Visp to Gornergrat via Zermatt takes about 2 hours.
Parking:
There are parking possibilities at the railway stations of Visp and Täsch (Matterhorn Terminal).
Map of the Zermatt region with public transports. (click to enlarge)
For further maps and timetables see the Homepage of the Gornergrat Bahn or the Swiss Federal Railways.