Pyhä-Luosto is accessible by car, bus, or taxi from both the E75 and the E63 highways in Lapland. The E75 goes from Rovaniemi to Utsjoki and you access the Luosto side of the park south of Sodankyla. From Kemijarvi, the Pyhä side of the park is on a turnoff about 30km up the E63.
Overnight train from the south arrive to both Rovaniemi or Kemijarvi, and then you can travel by bus from either of these two stations (Kemijarvi is closer). You can see bus timetables and buy tickets and on the Matkahuolto website
The closest airport to Pyhä-Luosto is Rovaniemi (RVN)