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The largest airport close to Malmo (particularly for international travelers) is Copenhagen Kastrup International Airport. here is direct train service from CPH to Malmo, about every 20 minutes. The fare (as of April 2014) is 110 Swedish Kroners. The station is built into the main airport terminal, there are directional signs. The trip takes about a half hour. Note that there are intermediate stops. Some trains will continue past Malmo. In addition many connections can be made at Malmo to other places. The Malmo main train station is adjacent to the downtown area, aswell as the train station Triangeln.
The Malmo Airport http://www.swedavia.se/malmo not only has daily connections with Stocklholm's Arlanda Airport, but also few international carriers and Hungarian low-cost airlines Wizz Air operating. Once at the Malmo airport, transport into the centre of the city is only about a half an hour. Upon arrival at the airport you can take a shuttle bus into the city. Shuttle rides cost about 99 Swedish kroners; for currency conversions check www.forex.se. The bus tickets can be purchased online, in self-service machines and information desk at the airport.
Taxis are also readily available, as with everywhere else, settle the cost of your ride into the city before taking off.