It's a 1.5-2h drive from the airport to Girona. Definitely rent a car, the taxi is super expensive and there's a lot to see on the Costa Brava.
While I didn't bring my bike with me I've visited the area about 4 times. Here's some things to see:

http://www.explorecatalunya.com/

Just read the description of one of the Costa Brava tours and you'll get plenty of ideas. We usually fly into Barcelona, rent a car and stay in the small towns along the coast (Blanes, Tossa del Mar). From there we take small day trips and we usually follow the itineraries of commercial tours on ouw own, without paying for them of course 😀

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What's your CdA?

Personally I would probably take a rental car, but train connections from Barcelona Sants to Girona are excellent (~1h). Really depends on how much you need a car in Girona.

Trains are dirt cheap.

If you take a taxi from the Barcelon Airport to Sants train station is is quite cheap for a high speed train ticket that takes you to Girona. Once in the train station nothing in the main city is that far away. Our experience renting a car in Spain was really quite expensive compared to North America. Also be prepared to accept a standard transmission since that is the general type of compact car. Around Girona it is really easy to go on foot and bike. We did rent and go the beach though we did it in May and the only people on the beach were us and about 20 other brave, cold tolerant souls. We tried to drive to Andorra but once we hit the high mountains decided it did not look like fun. So the car was a nice add on but is definitely not necessary. You can rent cars right at the train station in Girona. The times they are open though is strange.

I rented a bike from Christian et al at Service Course they are great people and can set you up well. They also provide you with a Wahoo that is loaded with great ride files. Depending on your style of riding they can give you some advice as well. Plus some great coffee! Definitely do Els Angels, it is so close to town and such a great climb. stop on the other side at Madremanya it is a really cool little town. As part of a nice 70 km route go up over Els Angels through to Monells then back over Santa Pellaia.

Another good loop is Mas Llunes. I did all of these solo using the GPS provided.

There is also a ride the leaves from the Bikes Breaks shop every Thursday morning. It is well worth the time to go with that group. They are very welcoming. Nice group and they break into different pace/ distance groups after the coffee stop. The other shop which I think has a ride Eat Sleep Cycle. I found the Service Course a rather high speed group and if that is your thing then maybe check out their Sunday group ride.

Simply put there are so many great rides to try you can not possibly do them all in such a short trip or maybe even in a year of cycling. You can cycle from Girona to the Mediterranean and back in an about 100 km route, Banyoles is a really nice ride when you get to the lake but the road to and from is pretty much highways etc.

You are in the foot hills of the Pyrenees mountains so in any direction there are some nice climbs. The local pro favourite is Rocacorba. It is a really tough climb with a dead end so it is up and back on the same road. The downhill I was told is challenging though I did not ride it.

Believe me Girona is one of the best places I have ridden and the drivers are very considerate. You will have a blast. As for family things I can not really comment on that because that was not part of our planning but the cycling is top notch and the Service Course is fantastic.