If you have only 2 days in Barcelona, this is what you can do.
Take the hop-on, hop-off bus called the Bus Turistic. The 2 day pass costs 23 Euros. unlimited rides.
Here's the link. You can check the bus schedules etc.
http://www.tmb.net/en_US/turistes/busturistic/rutesiparades.jsp
One day, do the Northern Route in RED color.
The other day, do the Southern Route in BLUE color.
The important places on the Northern Route are :
Sagrada Familia
Park Guell
La Pedrera
Casa Battlio
Futbol Club Barcelona (for those die hard Barca fans)
All these places except Park Guell and probably FC Barcelona (I didn't go there) have an entry fee. You get a discount if you show your Bus Turistic 2 day ticket/pass.
In the Southern Route, the interesting places are :
Plaça d'Espanya
MNAC (if possible go there on a Saturday evening. beautiful fountains at 7 PM on Thu/Fri/Sat evenings only. I missed it because I went on a Sunday)
Olympic Stadium (you can walk from MNAC to this place instead of waiting for the bus - just 1 stop away)
Barri Gotic (plan to spend at least 1/2 day here. beautiful architecture. i didn't have enough time for this...)
What I would recommend you do is that you do one round of each route without getting off the bus. Each trip will take 2 hrs. Keep marking the places that interest you. During the 2nd round, get off at those places.
At night, go to Las Ramblas. Take some pics of the graphic artists who use aerosol paint for their artwork. Its fantastic. Please don't go too late at night. Around midnight, it didn't feel too safe to walk alone - quite a bit of rowdy elements.
Don't rely too much on the Barcelona metro. Its the worst i've encountered.
Note that there is a train from the airport (outside Terminal A, its quite a long walk to the train station from the airport building) that runs every 30 minutes, into the city.
Have fun. I think Barcelona is a nice place, but wasn't too great an experience for me.