The obvious way is to hire a car and there are many places where you can do this. I have never driven in Spain, partly because I have seen how they drive in Barcelona (traffic signals do not seem to mean the same as they do in the UK) and also because it means that I would have to cut down on the cocktails, and after all, I am on holiday! Unless you are staying in a villa away from the main tourist areas, I really don't think it is worth hiring a car.
If you just need to get from airport to resort, you can book a transfer with one of several transfer companies. The upside of this is that if you are unsure as to where you are going you don't have to worry, as long as you can find your coach, then all is good as you will be dropped off and subsequently picked up outside your hotel.
The downside is that, unless you pay for a private transfer, which can cost £200 or more, you are working to someone else's timetable. I have sat on a coach, waiting for other passengers, for up to 2 hours before we have set off.
I have also struck lucky too, one year I booked a coach transfer, but as I was the only person, I was sent a taxi! A coach transfer is generally about £20 per person for a return trip, but this obviously varies by company and resort.
There are also public buses which run from the airport to Barcelona, it costs about €6 for a ticket to Placa de Catalunya in the city centre. From there you can find your hotel, if you are staying in Barcelona, or catch another bus or a train, or the underground to your end destination.
Another option is the train, which is nothing like in the UK, they tend to be on time and clean! Much like at Birmingham airport, you can catch the train from Barcelona El Prat with leaving the aiport, you just need to follow the signs along the walkway. All the trains go through Barcelona Sants station where you can change for pretty much anywhere you wish to continue on to!
Yes, that is a double decker train!
Travelling from Gerona, the best option would be to either book a transfer or hire a car, otherwise it is a taxi ride to the nearest station, about 4 km away, or a bus into Gerona itself and the train from the station there.
Barcelona is well supplied with dedicated cycle routes, some running alongside the roads, others completely separate. If you do not have a bike, you can hire them around the city. For anyone who wishes pay an annual fee (about €30), there are locations around the city run by Bicing where a bike can be picked up, used for upto 2 hours, and then dropped off at another location, you will see these all over the city.
For something a bit different, you can tour the city with Go Car. These are small yellow bug like cars which are driven like motorcycles, you drive round following the guided audio tour, an all day tour costs about £50 although there are other options available.