Apart from Barcelona, I would recommend la Costa Brava, especially from Sant Feliu de Guíxols up to north in El port de la selva. I highly recommend the hotel La Gavina in S'agaró. The area around Begur / far de Sant Sebastià is just spectacular. Cadaqués, too. Try to do the "camí the ronda" as much as you can. Don't stay on the same hotel more than one / two nights, change from location throughout la Costa Brava.
I also recommend the Pyrenees, especially the Natural Park Aigüestortes i el llac de Sant Maurici, for the people that like nature and mountains. It is just spectacular, and most people do not believe in Catalonia there can be such a thing.
If you are Christian and a bit religious, I also recommend a visit to Montserrat. It can be done from Barcelona, without spending the night outside Barcelona.
And for sure, if you go at the end of June (the best time of the year in Catalonia), try to be before Sant Joan (Saint John) approximately on June 23rd. There are big parties and fireworks overall in Catalonia, especially in Barcelona. A special night worth the visit, in order to mingle with locals: try "coca"!
In Barcelona, apart from the typical touristic visits, go to the parks in Montjuïc for half a day.