Since it's less than ten miles south of the city itself, most visitors to Atlanta fly into Hartsfield-Jackson Atlanta International Airport (ATL). This is a destination for Atlanta cheap flights from all over the world as well as lots of domestic American air traffic. It's well laid out with people movers to get you around and lots of interesting art exhibits to absorb as you wait for you flight. There's an extensive duty free shopping complex and several restaurants.
Hartsfield-Jackson Atlanta International Airport (ATL) has a train station in the main terminal building. Tickets are very cheap and it takes approximately fifteen minutes to reach downtown Atlanta. There are shuttle buses available but they're actually quite a bit more expensive. Taxis can be found outside the Arrivals area (look for the signs) and cost less than $30 even for a group, so they can be a practical option if you have a lot of luggage. It's also possible to book a limousine, and car rental facilities are available.
Most major airlines fly to Hartsfield-Jackson Atlanta International Airport offer Atlanta airfares, so you'll usually be able to find what you're looking for, with plenty of cheap flight options available especially if you book well in advance or book online. If, for any reason, you can't find what you're looking for, there are other options. DeKalb-Peachtree Airport is just fifteen miles to the north east in Chamblee, an easy trip by bus or taxi. It has modest facilities but is efficient and easy to navigate. Also nearby is Fulton County Airport, known locally as (Charlie) Brown Field, which also has basic facilities but has good cafés where you can wait for a taxi. Brown Field is a popular destination for specially chartered flights and is conveniently close to the interstate.
With so much choice, you'll find flying to Atlanta easy to organize and pleasant to do, leaving you more time and energy with which to enjoy the rest of your trip.