Zermatt is car-free and can only be reached by train. The standard route is by rail to Visp or Brig (easily reached from Geneva, Zurich, or Lausanne on Swiss Federal Railways), then change to the Matterhorn Gotthard Bahn narrow-gauge line for the 35-minute climb to Zermatt. The journey from Geneva takes about 3 hours; from Zurich about 3.5 hours.
If travelling by car, park at Täsch — the last village before Zermatt. A frequent shuttle train connects Täsch to Zermatt in 12 minutes. Parking in Täsch costs around CHF 15 per day. There is no road access to Zermatt itself.
The Swiss Travel Pass covers rail travel including the Matterhorn Gotthard Bahn but not the mountain railways (Gornergrat, cable cars). If you plan to use multiple cable cars and mountain trains, day passes for the lift system are worth considering.