By air you fly to Hammerfest with the small aircraft of Widerøe (http://www.wideroe.no) from Tromsø. Due to a lot of oil traffic, getting a cheap seat is difficult.
There are a couple of buses a day to Alta (http://www.ffr.no), from where you can get a cheap flight to Oslo with SAS Braathens (http://www.sasbraathens.no) or Norwegian (http://www.norwegian.no).
There is also a catamaran to Alta most days of the week. (http://www.ffr.no). The tour is highly scenic and recommended.
The famous Hurtigruten calls at Hammerfest (http://www.hurtigruten.com). It's about five hours to Honningsvåg and Nordkapp, and 12 hours to Tromsø.
