I lived in Barca many moons ago. One of my favourite places but as has been said, it can be a bit shifty at night. Keep out of the side streets towards the sea end of La Rambla, keep your wits about you and you should be OK. Whatever you do, do not eat or drink on La Rambla. The food will be average at best and massively overpriced. I am happy to recommend you some nice eateries and bars that are a little off the beaten track but worth it. Let me know what type of food you are after and I will hook you up.
If you are flying with Ryanair there should be a shuttle bus to take you from Girona airport to the town centre. If not, others on your flight will be in the same situation so offer to share a cab to Placa Catalunya which is a fairly central location.
In terms of what to do, as has already been said, get a ticket for the tourist busses. They cover all major attractions for a reasonable rate and you can hop on and off for up to two days.
Must see:
Sagrada Familia- Just breathtaking architecture. You cannot visit the city and not visit this.
Park Guell- On a nice day the view over the city is lovely.
Montjuic & Olympic Park- Similar to Park Guell for the view, get the cable car down to the port if you fancy it. I refuse to on the grounds that if the cable broke it would be such a rubbish way to die.
Bar Tomas in Sarria- On the tourist bus, one of the stops is Sarria which is a quiet little suburb of Barcelona famous for cakes and Bar Tomas!. It's worth getting off the bus here to just retreat from the hustle and bustle of town. Bar Tomas has simply THE best patas bravas (fried potatos with Aioli and Paprika) in the world. If you go all that way and do not eat them you will regret it.
La Boqueria- The covered market just off La Rambla. Have a browse around for fresh cheap produce. Cheeses and chorizo's to bring home with you can be had here at great prices.
Bosc De Les Fades- A bar attached to the Wax Museum at the end of La Rambla. Really funky and surreal inside with surprising decor. You feel like you are inside a forest, very cool
Do the Camp Nou tour. The missus will actually probably really enjoy it, the size of the stadium is just immense. If she doesn't fancy it, there is a cafe outside that she can sit in while you do it.