Best train from Madrid airport to University of Salamanca

Search renfe.com which is the owner operator of all of the trains in Spain and pick the route that takes the least amount of time (that is always the best one no matter where you go).

If you are wanting to take the train to Salamanca the same day you arrive, you should buy the ticket then and not in advance. The people all speak English and very helpful at the airport.

Does the school not have some sort of foreign student office you could ask about transportation options?

There are two major train stations in Madrid, the one you'll want to use is to get to Salamanca is Chamartín (as opposed to Atocha).

Unfortunately, he is not going with school. His University here cancelled all study abroad, so he is going on his own. So far, it's an adventure and hopefully won't become an ordeal.

Carlos, thank you for the specific station. That was the piece I was having a hard time with. His address is Calle Los Hidalgos and the bus nearest La Alamedilla doesn't go there. He'll just have to walk! Very European; not like where we live!

https://www.salamanca-university.org/Useful-Info.htm

It's 2 hours 45 minutes from T1 at the airport to Salamanca by the Avanzibus; cost is about 26 euros:

https://booking.avanzabus.com/web/step1.php

On google streetview it looks like a 20-minute walk from the bus stop to c/ Los Hidalgos near the bullring
but I bet there's a bus that runs to the bullring circle or a lyft-equivalent ridesharing app.