Vienna is your main gateway. Vienna International Airport (VIE) has direct flights from most major cities in Europe, North America, and the Middle East, with Austrian Airlines and a good mix of low-cost carriers. If you're heading straight to the Alps, Salzburg and Innsbruck both have airports with seasonal and year-round European connections — Innsbruck especially picks up in winter when the ski crowd arrives. Munich airport, just across the German border, is worth considering too: it's a major hub with more long-haul options, and Salzburg is only ninety minutes away by train.
Austria's rail connections are excellent. ÖBB (Austrian Federal Railways) runs direct services from Munich, Zurich, Milan, Venice, Budapest, Prague, and Berlin, among others. The Nightjet sleeper trains are a civilized way to arrive from cities like Amsterdam, Hamburg, or Rome — you board in the evening and wake up in Vienna or Innsbruck. From most neighboring capitals, you're looking at four to eight hours by train, and the scenery through the Alps makes it time well spent.
Driving into Austria is straightforward from Germany, Switzerland, Italy, or any of its other neighbors. The motorway network connects seamlessly across borders. You will need a toll vignette (Vignette) for the motorways — buy one at a gas station near the border or online in advance. A few Alpine tunnels and passes charge separate tolls. In winter, snow chains or winter tires are mandatory on mountain roads, and some high passes close entirely from November through May.