This is probably the least expencive way to get there, but it takes some time. There are 3 possibilities.
1. Train from Oslo to Fauske NSB
train travelinformation
Further with trainbus to Skutvik and ferry to Svolvær.
Trainbus scedual:
Fauske
Skutvik Svolvær
(departure) (departure)
(arrival)
0930
1245
1445
1815
2315
0115
2. Train to Bodø. Further with the ship "Hurtigruten" to
Svolvær.
(Hutigruten departure Bodø: 15:00 and
arrival Svolvær 21:00).
3. Train to Narvik (through Sweden).
Further with speedferry (OVDS) to Svolvær.
Price: NOK 267,-
Narvik
Svolvær
15:00
18:30 mon-friday
12:00
16:00 sunday
If you are a student, discounts are availible.