AIRporter provides buses from Derry to both Belfast International and the City Airport. During the Summer the buses are going every 2 hours. I always use them when I fly from Belfast. They even wait up for you when your plane is late! Check their website for more information and address.
There's no problem whatsoever to get to and from Derry. Derry has got its own airport about 30mins drive from the city centre. There's frequent buses and trains from Belfast and Dublin at reasonable prices.
All of Derry City can be explored on foot. It's not really that big. A hire car is always a good idea when travelling in Ireland anyway. You can rent cars directly at Derry City Airport, or in the city itself. Taxis are a very cheap way to get around as well. The taxis here have got no meters, so you pay a fixed prices (most of the time anyway) to get from area to area.