Things you MUST do:
1. Spend at least half a day in the Gothic Quarter; roam the narrow streets; visit the Cathedral. Unforgettable.
2. Stroll on Las Ramblas... Then go onto Passeig de Gracia. This is the most fashionable and fancy street in Barcelona; very very beatiful.
3. Plaza Catalunya. Here, you MUST MUST MUST see the fountain. At night it features light show with music. Absolutely unforgettable. So do this at night.
4. Things you may want to skip: Gaudi park (i think that' what it called); the waterfront.
Barcelona is one of my favorite cities, enjoy it!