Of course if you come to Barcelona there are so many places that are a MUST. From here, you are within easy access to Parc Güell (from Gaudí) or CosmoCaixa, the science museum that we strongly recommend especially if you come with children.
You can go up to Tibidabo to enjoy the views from the top-top, there is a very interesting church, and you can walk to the Collserola Tower, built to improve the communications for the Barcelona Olympic Games in 1992.
The center is only 20-30 minutes away by public transport, so really not so far to be able to enjoy all the great architechture and culture the city has to offer. If you are tired in the evening and don’t wish to use public transport, a taxi is always a good option and not so expensive (normally between 10 and 20 EUR depending where you are and the time).