Travelling to Lapland is very easy and there are many options including bus, train, plane or your own vehicle.
The following map provides an overview of the Nordkalotten Trail between Kvikkjokk and Abisko.
There are a number of of options when travelling by train to lapland, I prefer the night trains, there are trains commencing in Göteborg, Malmö and Stockholm, all of which will get you to Lapland with a minimum of fuss, I used the Swedish rail booking system to book my ticket to Murjek.
From Murjek I caught the Länstrafiken Norrbotten bus, Linje 94, which passes through Jokkmokk before arriving at Kvikkjokk. There is a short stopover in Jokkmokk which gives you sufficient time to visit the nearby ICA store for those last minute supplies, as well there is an alternative rail service which may be of interest. Along with being the start of the Padjelantaleden and Nordkalotten, Kvikkjokk is also the start of the Kungsleden which will take you north to Abisko there is also a large STF Hostel in Kvikkjokk.
From Kvikkjokk the start of the Padjelantaleden and Nordkalotten can be accessed via boat a short walk from the bus terminus in Kvikkjokk.
If you are commencing hiking further north then the trail can be accessed at Ritsem. A bus service from Gällivare, Linje 93, will take you to the STF lodge at Ritsem, a ferry service (scroll down) then takes to the trail at Vajsaluokta, or you can head north from Ritsem towards Hukejaurestugan and then on to the Kungsleden.
Transport alternatives in Abisko include the night train as well as several regional trains to Kiruna and beyond to Boden, or alternatively train to the Norwegian port city of Narvik. I used the regional rail service to Kiruna from which I flew to Stockholm and then home to Copenhagen. The choice of flights to and from Kiruna is limited so booking early is recommended, especially during the peak tourist seasons.
There are also bus services linking Narvik and Kiruna with Abisko, information about these can be found at Länstrafiken Norrbotten.
Heading further north? In 2010 I commenced at Björkliden railway station (west of Abisko) and walked north to Kilpisjärvi, from where I caught a bus to Tromsø and flew home from there. The bus service was excellent an provided some stunning views of the Norwegian mountains before arriving in Tromsø.