You will find a comprehensive guide to travelling to and within Barcelona, here.
The Summer School will take place in the Faculty of History, University of Barcelona. Located in the very centre of the student quarter it is easily accessible via metro (stops: Universitat or Catalunya) and a five minutes walk from Plaça de Catalunya and the Ramblas.
Arriving by air
The best way of getting to the city centre from the airport is by bus. Aerobus runs on an every-five-minutes schedule and will take you directly to Plaça de Catalunya for a few euros. If you need to get to another part of Barcelona, the L9 metro line connects the airport to the metro network.
Arriving by train
If you arrive by train, it is likely you will end up at the Sans station. There are frequent local train connections between the station and Plaça de Catalunya. Alternatively, the L3 metro will take you directly to the city centre.
Arriving by bus
If you take the bus to Barcelona, chances are you will arrive at Barcelona Nord bus station. The L1 metro connects it with the centre.
Metro, buses, tramps
The best way to use public transport in Barcelona is to purchase the T-10 ticket. It gives you 10 journeys for roughly 1 euro each. It can be used on all types of public transport in the city and you can use it for multiple people at the same time (by validating the same ticket once per each person). You can buy T-10 in all metro stations.
You can find the map of Barcelona metro here. The city operates a network of buses crossing the city in a grid fashion (V-buses that go vertically: north-south and H-buses that go horizontally: east-west). You can find the map of these bus connections here. There is also a dense network of local buses. It may be worth checking on google maps what is the quickest option for you as the metro is not always the best choice. The T-10 ticket works on all of the described modes of transport.