Gällivare is along the "Malmbanan" iron ore railway which runs between Luleå on the Baltic Sea coast, and Narvik on the Norwegian coast. This means railways up here are a tradition since navvies sacrificed their lives to create the line that would send Swedish iron ore across the world. There has been night trains to Gällivare (and on to Kiruna/Narvik) from Stockholm for ages and the 16 hours on a train that this takes is actually great fun as you have all sorts of accommodation to chose from and there is a buffet car with food, drinks and breakfast deals. There is constant light if you travel in summer, so that you can see the majestic rivers and desolate cottages you whizz by, whilst in winter, you wake up to Arctic winter light and reindeer outside your window. There is a threat to terminate the night trains in Luleå in future which I hope never happens but in that case you will have to change trains in nearby Boden which you might still have to do on some departures.
Gällivare station is a wonderful wooden creation from 1893-94 in a national romantic style and you can really see the importance of the railway up here since the size of it is bigger than the size of the town merits.
Airplane is the fastest and most comfortable way. You can go by train and bus as well but I think it takes to long.
Walk! Gällivare is not a big town. But if you like nature but don't like long walks, rent a car.
The Norrland way to transport yourself around town in wintertime is by "spark" which translates as "kick" as that is how you move forward on the snow.