I suggest you take a trip to Girona: very nice city, and it's only 1 hour away from Barcelona by train. It costs around 5 euros 1 way, cheaper than a bus. There's an amazing gothic cathedral and the jewish quarter is great for a walk. Very safe city.
In Barcelona you could spend some time in the Gothic Quarter ( cathedral and its surroundings ) or see the modernism of Antoni Gaudí in Passeig de Gràcia: you got buildings like Casa Milà or Casa Batlló. You can get there from the main Square, Plaça Catalunya. You can get the best view of the city from Park Güell: modernist park also designed by Antoni Gaudí. Metro line 3 - green "Lesseps" stop and it's a 20 minutes walk up a hill. Montjuïc Palace is amazing, with its fountaines in front of it: metro line 1 - red "Plaça Espanya" stop.
But you wanna see medieval things...I don't think Barcelona is the right spot for you then to be honest. This city is known for being a modernist place. The more medieval you'll go in Barcelona is the gothic Cathedral, unless i'm forgetting some place.