There are 4 popular locations to start/end the Tour of Monte Rosa - Saas Fee, Zermatt and Grächen in Switzerland, and Alagna in Italy. Learn more about these locations in this detailed blog post. There are two main methods for getting to/from the trailheads; public transport and driving.
Public Transport
If you are starting your hike in Switzerland (Saas Fee, Zermatt or Grächen), travel time from all 3 airports are between 3.5 and 5 hours by public transport. Trains/buses depart once or twice per hour and will cost between 85-110 CHF. See SBB.CH for schedules, prices and online ticket purchases.
If you are starting your hike in Italy (Alagna Valsesia), you can catch a combination of buses and trains to reach Alagna. Options include train and bus via Novara, or buses via Milan. The frequency is varied by day and time. Approximate travel time is 5 hours and will cost between 20-35 EUR. Rome2Rio provides a good overview of the transport options by day, time and type. Some trains and bus tickets can be pre-purchased.
2. CAR HIRE
If you are travelling as a couple or with a group, hiring a car can be more affordable than public transport (especially in Switzerland) and is typically faster than public transport (especially for Milan > Alagna Valsesia). Car hire cost varies depending on when you book, pick up location and type of car. Rentalcars.com provides a good variety of options by budget and car size. Costs from Geneva & Milan airport starts from 40 EUR / day.
Destination Saas Fee: Travel time from Geneva Airport is 2.75hrs (v. 3.5hrs+ with train and bus). Upon arrival in Saas Fee, there is a large covered and uncovered parking lot. Cost of parking is CHF 136.50 for 2 weeks or ~10 CHF per day).
Destination Zermatt: It is not possible to drive all the way to Zermatt as it is a car-free village. Therefore you must park your car at the village of Tasch and then catch the train to Zermatt. Travel time from Geneva Airport to Tasch is 2.5hrs and Tasch to Zermatt via train is 10 mins. There are several parking lots available. The Matterhorn Terminal charges CHF 203 for 2 weeks or CHF 16 per day. Train ticket (one way) to Zermatt is 8.20 CHF and can be purchased online or at the train station.
Destination Alagna Valsesia: Travel time is 1.75hrs - significantly less than public transport. There is free parking available on the streets within Alagna Valsesia, as well as paid parking options (EUR 7 per day).