There are four "types" of trains going to Salzburg, the regional BRB, the Austrian Railjet, Westbahn (a private rail company), and a Bahn long distance train (Eurocity, EC, or InterCity Express, ICE). None of these trains leaves from the airport; it is necessary to take an S-Bahn from the airport to either the Hbf (main station) or Ostbahnhof (east train station). All of the trains stop at the Hbf. The Bahn long distance trains are coming from elsewhere in Germany and stop for a few minutes at the Hbf; the others all originate at the Hbf. All but the Railjet stop at Ostbahnhof. You have to go to the Hbf (usually on S1) to catch the Railjet. For the other three, it's faster to take the S8 to Ost and catch the train there, but you can also catch any of the trains at the Hbf.
All of the connections, Railjet, connecting at the Hbf, and the others connecting at Ost, take about the same amount of time, airport to Salzburg Hbf, 2H15 for the EC, 2H18 for the BRB, and 2H27 for the Railjet, so it doesn't really matter, timewise, which connection you take, but the BRB runs hourly, and the others every two hours.
Since the Westbahn is a private company, the Bahn does not sell tickets for it. You'll have to go to their website (or, there might be a Westbahn ticket counter at the airport). Westbahn ticket prices seem to vary a little by date and time of day, but they are around 30€ for Ostbahnhof to Salzburg. Westbahn tickets do not include the S-Bahn; you'll have to buy that ticket (13,60€) separately.
The BRB, using a Bayern-Ticket is 29€ for one person (10€ for each additional person up to 4 additional). The ICE and Railjet tickets (the Railjet is considered an ICE) are about 48€ for each person and could be used on any train Bahn train, the EC or BRB. The EC ticket is about 44€ per person and can also be used on the BRB but not the ICE/RJ. The prices I'm giving for the non-BRB trains are for a fully flexible ticket, valid all day long. I would not recommend a SparPreis ticket for use right after arriving on a trans-Atlantic flight.
The S-8 leaves the airport for Ost at 4, 24, and 44 minutes past the hour. If you are taking the BRB and leave on the :24 S-8, you'll have 11 minutes to catch the train at Ost. That should not be a problem. You go down the stairs from where you get off the S-Bahn and walk about 180 ft. to the third stairway from the one you came down, to platform (Gleis) 8. Nominally, the train stops for only one minute at Ost, but if you are in the line to board the train, the conductor is not going to release it until you are on board. And, depending on when the plane arrives and you get through immigration, you can always take an earlier S-Bahn and have more time at Ost.
Here is the plan for Ostbahnhof. The S-8 from the airport will arrive on platform 1.