You can get to Abisko pretty easily by train; it’s just before the Norwegian border on the Kiruna–Narvik run. The Inlandsbanan will get you to Jokkmokk, from where you can get a bus to Kvikkjokk, another point on the trail.
There are also several useful bus routes that you can take to link up with the trail, listed below; most of these services are run by Länstrafiken Norrbotten and Länstrafiken Västerbotten (w ltnbd.se and w tabussen.nu). The buses operate a bussgods service, which allows you to send your pack ahead to your destination or, alternatively, back to your starting point, sparing you the effort of lugging your stuff around; ask about this service at bus stations or on the bus.
#31 Hemavan to Umeå
#47 Jokkmokk to Kvikkjokk
#92 Kiruna to Nikkaluokta (19km from Kebnekaise fell station)
#93 Gällivare to Ritsem (passes through Vakkotavare and Kebnats for the boat to Saltoluokta)
#200 Arjeplog to Jäkkvik
#341 Ammarnäs to Sorsele