Take the Barcelona Bus Turístic and explore other parts of Barcelona outside the center is one of our recommendations!
In the article "The Top Things to Do in Barcelona (and costs!)" posted to the blog Mapping Barcelona I list the top places you can't miss.
You can buy a Barcelona Bus Turístic 1 or 2 day ticket which will allow you to explore the city at your leisure. Some suggested stops (in addition to all the ones I’ve listed): Montjuïc (a hill with a funicular and great views overlooking the city.
Holds the National Museum of Catalan Art, Tibidabo (for a church, a fairground, and more incredible views of the city), Parc de la Ciutadella (huge park with lakes, monuments and the Barcelona zoo), Barceloneta beach (the most famous of many city beaches), Camp Nou (the football stadium of FC Barcelona and one of Europe’s best football stadiums).
Besides helping you get to specific sites of interest, the hop on/hop off is always a great way to see a city.