At the moment you just got one commercial airfield in Gothenburg
But there is, actually, a couple of flights to this little airport too. And a lot of possibilities to buy affordable air tickets.
landvetter gothenburg airport
40 km east from gothenburg city. You've got wifi where you are able to check your email with laptop, you've got restaurants, cafés with fresh swedish coffee open 24/7, Forex for currency, shopping, storage boxes for luggage, Disability Services, Lost Property service that you reach by phone +46 31 94 14 30, hotel and everything you need. To get from the airport to Gothenburg the easiest and most affordable way is simply to use the airportbus. With airportbus you'll reach city by half an hour. They charge 70kr, 5 £, 8€ or 9$.
New lines and flights has opened during this summer. Icelandic and Norwegian air are both opening new flights. Norwegian to Malaga, Nice, Pristina, Palma and Chania. Icelandic a new flight to Reyjkavik. You are able to get from Gatwick in London to Gothenburg as low as 21$.
The old city airfield is now just used by smaller airplanes. You also got a museum for older planes. The museum is called aeroseum and you'll find more about it on aeroseum.se
onetravel.com is another page that are finding the best deals for flying. Check this and the flightnetwork for finding the best price. The difference here is that onetravel gives a discount of 10$ or more.
On radarvirtuel you are able to see where your airplane are currently. Click here>>>
Radarvirutel is a kind of googlemap with all aeroplanes in the air and you can see what kind of plane, flying, where it is going and a lot of other stuff by clicking on it.
Flight buses
To Landvetter you actually got three different buslines. The most comfortable and with most tours is flygbussarna.se where you pay around 10$/€ one tour. You are also able to take 
Swebus or nettbus to the airport from the Nils ericsson therminal in Gothenburg(near the central station.
They want about 7 € or $ instead. Check it out on google maps>>>. 
You also got bus 612 from Landvetter centrum two times an hour. You see the timetable on vasttrafik.se>>>.
Another way could be there use bus 611 to Harryda kyrka. Then you'll get a kilometer or two up to the airport.
Could be a good excercise before flight if you are flying for a long time. If you start in Gothenburg center you'll need to go with red express to Landvetter travel center.
The time table you'll find here>>>
Affordable tickets to and from Sweden