Salzburg Airport: Skip private transfers. Take the airport bus to Salzburg Hbf, then ÖBB trains via Attnang-Puchheim to Bad Aussee; allow about 3 hours from the airport, usually with one change after Salzburg. Budget roughly €25–45 booked close in, less with an advance Sparschiene fare. The rail run after Attnang-Puchheim is part of the point: lakes, mountains and the Traun valley.
Salzburg Hauptbahnhof: The sensible rail route is via Attnang-Puchheim, changing for the Stainach-Irdning service that stops at Bad Aussee. Allow about 2½ hours; fares are commonly in the €20–40 range depending on ticket and timing. A car is only faster if you already have one and are not travelling at a busy weekend.
Graz Airport: Take the S-Bahn or bus into Graz Hbf, then an IC westbound to Stainach-Irdning and the regional train to Bad Aussee. Allow roughly 2½–3 hours airport-to-town and about €25–45. This is the best public-transport gateway from southern/eastern Austria, not Salzburg.
Vienna International Airport: Take an ÖBB Railjet or S-Bahn into Wien Hbf, then use an ÖBB itinerary via Leoben and Stainach-Irdning, or occasionally via Attnang-Puchheim. Expect roughly 4½–5½ hours door to door and about €35–75, depending heavily on advance purchase. Do not pay for a long private airport transfer unless several people are splitting it.
Bad Aussee station: The station is just outside the compact centre. A local bus takes about 3 minutes when it lines up; otherwise walk, allowing roughly 20 minutes with luggage. Check the bus timetable rather than assuming frequent urban service.