Try to find a flight to Lyon, thats the closest international Airport to 3 vallée (be sure you search for the right 3 vallée, including the name of the town
, there's another place far west in France called like that). From there, it's about 3-4 hrs drive to Verbier and 6-7 hrs to St.Anton in Austria. You'll need to drive through entire Switzerland to get from 3 vallée/Verbier to Austria, most of the time you'll be driving on the A1, the traffic aorta of our country. Try to avoid the stretches between Geneva and Lausanne, and between Bern and Zurich during the week between 4pm and 6pm and on Saturdays, these are the traffic jam sections the little 170km can take you 3hrs+ then.
Rent a car with Europcar, they're by far the cheapest I know of.
Youthhostels in Switzerland cost about 40 $p.P. for a 3 bed room. Closest to Verbier would be Sion, a bit further away Montreux (yes, the smoke on the water Montreux).
Sion - Hostels - Swiss Youth Hostels
Switzerland is by far the most expensive country of the three, but also here you can easily stuff 3 guys with 50 $. You won't eat in restaurants, but loaf of bred 3 $ and a pack of 5 (very nice!) Cervelats sausages 5 $ will serve as dinner. Don't eat on the mountain, the food there is insanely expensive. Bring your sandwiches and beer. If you're not fussy with beer, you can buy a half liter can for 1$. Chocolate is very cheep here and has a lot of fat (I lived many years from chocolate, bread, pasta and sausages for 6 $ a day). In France, however, nutrition is very cheep, about half the price. There you gonna get beef for the price of sausages here
.
A liter gasoline costs 2$. Diesel is even more expensive, so don't rent a diesel car. Be aware, that the French highways are private, so they charge you for using them. My father drives theway from Lyon to Geneva very frequently, so if you want, I can ask him for the exact price of this stretch. Highways in Switzerland are for free, only thing you need is a "Vignette" (40Fr. for 1 year) but I assume, the rental car will have one. If not, you'd need to buy it at the frontier.
Other possibility would be to fly to Zurich and rent the car there. Zurich would be in the middle of your three destinations. Depends on which resort you're going to ride first... Zurich is 120miles from St. Anton and 180miles from Verbier. Guess, on US scale, that's close together.
Er, and be aware, Switzerland is not part of the EU. Own currency. They'll accept Euros in resorts and motorway service areas, but for a bad exchange rate.
Wether or not you rent the car in Switzerland, do the booking over europcar.com and not .ch! Saves you again some bucks (strong Fr.).