Sadly 2 days will only scratch the surface.
I have been 4 times and would go again. Last time I had a street map and armed with plenty of water I walked 15 miles taking in so may places i.e. The Nou Camp.
Sagrada Familia............as others say........that is a must see. If you want to go inside for the tour make sure you are early early early otherwise you will be queuing for hours.
A LEASURELY walk down The Ramblas.......take the time to digest it. Worth buying a beer/coffee and sitting in the middle to people watch. The beer is ridiculously expensive but IMO its worth doing.
If you (or anyone you know) is football fan you can buy excellent replica Barcelona football shirts. They were really cheap but the may not be nowadays.
There is a fabulous church just off Ramblas (to the left, near the top as you face the sea).
Near the church there is a great Tapas bar.........they pass small tasty tapas around and each one has a different coloured plastic sword in. You take what you want and collect the swords......they know exactly how much you have 'spent'.
@bianchi1 hinted

.............watch those b***y pick pockets THEY ARE EXPERTS. Don't kid yourself that you wont get caught.......keep your wallet/passport in a place they CANNOT get at.
Enjoy the 'human statues' as you walk the Ramblas but be aware.........if you pause too long and especially if you take a photo.....they will expect you to chuck some money in the hat. IMO most of them deserve it as they are so good and it must be really hard work.
Enjoy yourself as I am now thoroughly jealous
