If you ask people in the street for directions, they will most likely not know "Residència d'Investigadors", since the Residència is only open to people related to one of the Universities or research centers in Barcelona. Non-tourists (there are a few of those on the Ramblas) will know, though, "carrer Hospital" (the street where the Residència is located).

You can take a cab. Barcelona cabs are black and yellow (very easy to spot). They are parked just outside each of the airport terminals. The trip will cost you some 30-35 euros plus a minor charge per bag. Cab drivers do not expect tips but, if you wish, a small tip  [1 to 1.5 Euro] is appropriate. For Americans: in general, tips should be about half, or less, than in the US.   Give the cab driver the following address: Carrer Hospital, 64.

By train. There are signs at the airport indicating the way to the airport train station (the company that runs the train is Renfe). All trains that stop at the airport have a stop at  Barcelona-Passeig de Gracia (this is the stop after Barcelona-Sants) . Once in Passeig de Gracia, take the metro (underground) line 3 (the green line) direction Zona Universitaria to ‘Liceu’. From there it’s five minutes to the Residència.

By special bus. The "Aerobus" runs between the Barcelona Airport and Catalunya Square ("Plaça Catalunya"). It stops in front of each of the airport terminals. It costs  4.25 euros. The trip takes 30-40 min. There is a bus every 6-15 minutes, depending on the time of the day. You purchase the ticket on the bus. It has four stops: Plaça Catalunya is the last one. 
From Plaça Catalunya you can reach the Residència either
(i) by walk: 10-15 minute nice walk down Les Rambles;
(ii) by metro take the green line in Plaça Catalunya, direction Zona Universitaria. It is just one station: you should get off at Liceu. Then it is a five minute walk;
(iii) by taxi: it might cost some 4 euros.