From both Zurich and Geneva airports, Zermatt is just over 3 hours by train. By car, Zermatt is also around 3 hours from Geneva, and 4 hours from Zurich.
Zermatt is car-free and is closed to private traffic. Electric-cars (including taxis) and horse-drawn sleighs and carriages are the only transport allowed.
If you are arriving by car, you simply park at Täsch (the next village down from Zermatt) and the regular shuttle train brings you to Zermatt in a quick 10 minute trip. Alternatively, hop in a taxi.