Ah love that city 😀
La Acadamia for dinner is great but get there when they open or you'll be waiting for at least an hour for a table. Thats tucked away in the maze of tiny streets in the Barie Gothic area well away from the main tourist thoroughfares.
Almost next door to La Acadamia is a little bar called Ginger which has a really chilled vibe and plays some soothing beats while serving great cocktails.
For Paella try Set Portes down by the harbour (near the parliment building) for lunch its a great place to people watch while eating great food, the great and the good of Barca can normally be found there.
Have a look at the Time Out guide to food for Barcelona as thats got some other places that might be a bit nearer where your staying.
The Gaudi park is called Parc Guile (sp) and is a great place to while away a sunny morning or afternoon. Make sure you check out the Salvidor Dali house there.