A: First of all, it is important to know that there are two different venues of the Fira de Barcelona's exhibition centre close to each other in the Montjuic area. One is the Gran Via Venue, the other one the Montjuic Venue.  There are also other Fira centres around Barcelona therefore it is important you are clear on the Fira centre you are going to. 

From the airport, there is currently no direct transfer to reach the Gran Via exhibition centre by public transport. For some trade fairs the Fira Barcelona offers a free of charge bus service leaving from the airport, but not for all of them. The easiest way is to transfer from the airport now is to catch a taxi at the airport which brings you directly to the venue. Otherwise you can reach it by combining different public transportation means explained below. There are two main options. First you travel from the airport to Placa Espanya and then travel by bus or FGC to the Gran Via Fira Centre or the second option is to travel to Estacio de Sants using the RENFE train from the airport and then catch the metro to Placa Espanya where you can catch the FGC or bus.

Aerobus to Placa Espanya
You can catch the Aerobus (airport shuttle bus) from Terminal 1 or 2 and get off at Placa Espanya once at Placa Espanya you can use the Train or Bus to arrive at the Gran Via Fira Centre.

Train from Placa Espanya to Gran Via Fira centre
There take the Ferrocarrils de la Generalitat de Catalunya (FGC) to the stop FGC Europa/ Fira (please find the exact details of the different lines below).
Travel by Ferrocarrils de la Generalitat de Catalunya (FGC) from Placa Espanya to Gran Via Exhibition Centre:
The nearest stop to the Gran Via exhibition site is the station "Europa/Fira". All FGC trains that leave or arrive at the "Plaça Espanya" FGC station stop at "Europa/Fira", the third stop. The FGC trains that stop at "Europa/Fira" are
 FGC trains from Placa Espanya direction Martorell-enllaç
 FGC trains from Placa Espanya direction Manresa
 FGC trains from Placa Espanya direction Can Ros
 FGC trains from Placa Espanya direction Olesa
 FGC trains from Placa Espanya direction Igualada

Bus From Placa Espanya to Gran Via Exhibition Centre
Another possibility, also leaving from Placa Espanya, is to take the bus 79. The bus stop near Placa Espanya is located at Avenida Paral.lel, one of the main roads that leads to Placa Espanya. The bus goes in the direction to the Metro station Carrilet. You have to get off at the stop Calle Botanica/ Salvador Espriu. This is the bus with the nearest stop to the Gran Via Exhibition Centre.

Travel by metro from Sants Estacio railway station to Gran Via Exhibition Centre:
You can also take the RENFE train from the airport and you would then get off at "Sants Estacio" railway station. There you have to take the metro   Metro: Sants Estacio (Green Line, L3), in the direction of Trinitat Nova. The entrance to the metro is inside the Sants train station itself. Just follow the signs for the metro. Once you have caught the metro at Barcelona Sants you then have to get off at   Metro: Espanya and there catch the FGC or bus as indicated above to arrive at the Gran Via Centre.

The exact addresses of the Fira Gran Via Venue and Montjuic Venue are:
Gran Via Fira Centre
Carrer Botànica, 62
08908
Hospitalet
Barcelona

Montjuïc Fira Centre
Avenida Reina Maria Cristina s/n
08004
Barcelona 

Note: s/n in the address above means "sin número" in Spanish which means no number. There is no street number on this building.

You can find more information about the exact location and how to arrive to the venues by clicking on the link of the official Fira de Barcelona's website indicated below:

Official site Fira de Barcelona

Transport from Barcelona Airport El Prat

Taxi rates in Barcelona