You can get from the airport to the centre on the RENFE train, fairly cheap and regular (although the taxi will take no more than 20 mins in low traffic and they mostly take credit cards and would be about 30 euros). I think if you do this you need a T10 ticket, this entitles you to use the terminal bus also.
The tourist buses are good and the hop on/hop off tour covers most of Barcelona, I highly recommend the Noucamp even if you are not a football fan, the stadium is an awesome site. Other must do's are: Sagrada Familia, Le Pedera and Las Ramblas - the aquarium and zoo are pretty good too but whatever floats your boat!
Do avoid the side streets, especially on Las Ramblas after about 8pm, Barcelona is a very safe city but it has some crime as all major cities do.
As for eating, most places are quite cheap, go for the cheap set menus for lunch and early supper which will save a fortune. If you only fancy a snack the supermarkets are good - there is a carrefor (I think) at the end of Las Ramblas (heading towards the sea) which is cheap when you don't want much.
If you get a chance take a few hours in Sitges, less than an hour on the RENFE and the monastery overlooking the bay is the most beautiful place I have ever been.
As for restaurants, I love El Convent on Calle Jerusalem, which serves traditional Spanish food at good prices (and always friendly to the children) and for a blowout the Posit de Pescadors overlooking the seafront is amazing, but only for special occasions.
M'encanta Barcelona 