Without doubt, the best way to get around Swedish Lappland is with your own set of wheels, whether that be a rental car, motorbike or bicycle.
The Inlandsbanan train runs in summer from Östersund in the south to Gällivare in the north. You'll need to use buses to explore outside of the towns it stops at.
The bus system is extensive. Länstrafiken Norrbotten (http://ltnbd.se) runs the buses in northern parts of Lappland, while Länstrafiken Västerbotten (www.tabussen.nu/lanstrafiken) does the same in southern parts of Lappland. Check the websites for up-to-date timetables and prices.