Sagrada Familia definitely. Book ahead and you'll get a fixed time. It costs extra to go up the lift (i think...).
On the first Monday of the month many museums are free. There are Picasso, Miro and Chocolate museums! Although I don't know which of those, if any, are free.
There are 3 tour buses, you can get tickets for one, two or all three for a day or for your entire visit. They stop everywhere you want to go, including awkward-to-get-to places and the Nou Camp stadium that houses Barça.
At the bottom of Las Ramblas is a statue of Christopher Columbus, you can go up him. Fab views.
There is a cable car and funicular going up Montjuic mountain from the port - the Olympic stuff is on the mountain. They used to do F1 up there too, until the early 90s.
While you're out and about, I always found the chain restaurants FresCo and Latuca to be bargainous: for about €10 each eat all you like buffet, 3 courses plus drink and coffee. There are several dotted about. Very fresh quality food.
I love walking up and down Rambla Catalunya (parallel to Passeig de Gracia- the big main street) as it is padestrianised and full of trees and bars and restaurants. If you're going up that street, turn left onto Mallorca street and just there on your left is a fantastic restaurant called Cerveceria Catalana (i'll double check that with DH!). Top quality food, medium price, tapas.
The Born district is lovely and well worth a wander (and get pizza from Pizzeria del Born on the main street Paseo del Born). Beautiful church: Santa Maria del Pi which book The Cathedral by the Sea was written about.
So much to do and see, it is a great city.