Regarding Dubrovnik, we stayed in the Lapad area to avoid some of the crowds of Old Town. The #6 bus will get you from Lapad to the Pile Gate of Old Town in just about 10 minutes. Lapad has a couple beaches and some decent restaurants with a little bit better prices than in Old Town. However, if you want to be in the middle of it all, stay close to Old Town.
As far as what to do, it's the standard stuff you'd see in any guide or web site: City Walls, Old Town, Cable Car, day trip to Montenegro, day trip to Bosnia. All are fantastic.
The airport is in the next town down the coastline because because there is no space for a runway next to the mountains of Dubrovnik, so just know that you'll have about a 20-minute taxi ride to Dubrovnik itself.