Copenhagen Airport (also known as "Kastrup"), in Denmark is a major hub for Northern Europe with over 26 million people using the airport in 2015. It is the closest international airport to Lund and the journey time by train from the airport to Lund is about 35 minutes. The journey takes you over the Öresund Bridge connecting Denmark to Sweden and via Malmö central station where you may need to change trains.
Train tickets can be purchased from the Skånetrafiken ticket machines in the arrivals hall at Copenhagen Airport, near the escalators going down to the train platform. You can pay using a debit/credit card, and if you buy a ticket all the way to Lund, the ticket will be valid even if you change trains at Malmö central station. You can also use the Danish DSB ticket machines when travelling with Skanetrafiken, or make your purchase at the DSB ticket office; however, a DSB ticket will be slightly more expensive.
The closest local airport to Lund is Malmö Sturup. It has some international flights from destinations within Europe and flights from other Swedish airports including Stockholm. There are bus connections to Sturup airport ('flygbus') or you can get a taxi to Lund.
Cycling is probably the most popular form of transport for journeys in Lund. The flat terrain and miles of dedicated cycle lanes mean the city is ideally suited to cycling.There are many rental shops throughout the city (see, for instance, http://www.begcykel.com/cyklar/city-bikes).