Many people are starting to find out that Girona is a pretty darn good base and visiting Barcelona is quick, easy and cheap. The best and most relaxing way is by train and there are regular trains every hour with the Express trains taking about an hour and a quarter. The only thing to be aware of is that the last train from Barcelona is at 9.20pm so don't miss it as it would mean either a taxi back or staying in a hotel overnight, and both options are expensive! The cost of the train is a very reasonable 12€ (approx) return which compared to the motorway tolls, petrol and parking is a real bargain.
The best stop in Barcelona is "Passeig de Gracia" which drops you off very close to two of the best Gaudi buildings the Casa Batllo and the Pedrera and once you have seen them (and visited the excellent shop in the bottom of the Pedrera for gifts) Plaça Catalunya and the Ramblas are an easy 5 min walk. The Ramblas are charming and crowded so watch your handbags/wallets and the living statues are rightly fascinating. If you walk all the way to the end you come out in the Port area where there is a good aquarium and an IMAX cinema as well as a food court and many shops.
I always recommend a loop so walk along the port until reaching Via Laetana off of which the Cathedral and various impressive government buildings can be seen along the narrow streets of the oldest part of Barcelona the "Barri Gothic." All roads lead back to the Plaça Catalunya and from there the train station is only a few minutes walk.
If you are interested in the local gastronomy then do not miss a walk around the Boqueria market (1 third of the way down the Ramblas on the right) and then instead of the plastic food at the port discover some of the wonderful bars off to the right of Laetana such as La Vinya del Senyor (for wine by the glass and food) or Cal Pep for fresh and simple cooking.
All in all you will have seen the main Barcelona sights in an easy day trip although there is still plenty left a bit further afield for the next time!