There are several trails along the Skåneleden Trail in Skåne (Scania), the southernmost province of Sweden. There is also good public transport. Skåneleden
Years ago I hiked three days and two nights on the trail leading Eastwards from Malmö.
Map
I took a bus from Malmo town to Torup, abt 30 minutes to start.
Bus Departure; Malmö C. Arrival; Torup
Check Google Maps for "Torups rekreationsområde" to get an idea of where the starting point Torup actually is at.
It is a nice hiking trail. It goes in lovely nature and passes several lakes of various size. There are no formal camping grounds that I can recall. But many sweet places for wild camping. Several spots have a simple shelter where you can wait out rainy weather. There is an outside place where you can safely make a fire for cooking, there is chopped up firewood and there is a bucket for drawing water from the lake.
Do it!