If you're coming from northern Italy, you can get to the Eastern Alps from Villach on the E55/A13 in Italy, which becomes the A2 and then the A10; from Klagenfurt, farther east in Carinthia, taking the A2 autobahn is quickest. The fastest route from Salzburg is the A10 autobahn, but you'll have to take two tunnels into account at a total cost of €11 (Tauern- and Katschbergtunnel). In summer on certain weekends (especially during Germany's official holidays), the A10 southbound can become one very long parking lot, with hour-long waits before the tunnels. Taking the normal road over the passes, although long, is a very attractive option, but you won't be the only person who thought of it. A good alternative to the busy Tauern Highway is the Tauern Motorail link, connecting Bad Gastein with Carinthia (Mallnitz) through an 8-km (5-mile) tunnel. It’s a 20-minute ride and cars are transported from one side to the other by train while passengers ride in a proper carriage. One way costs €17, round-trip €30. Prices are per car, including all passengers. If coming from abroad, don't forget to buy the autobahn vignette (sticker) for Austria .
Be aware that the Grossglockner High Alpine Highway is closed from the first heavy snow (mid-November or possibly earlier) to mid-May or early June. Though many of the other high mountain roads are kept open in winter, driving them is nevertheless tricky and you may even need wheel chains.