For connections along the coast and in the far north you have to go with buses. For example from Bodö/Fauske to Narvik, to the Lofoten or from Narvik to North Cape. Narvik is connected by train from Sweden.
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To travel from Norway to Sweden by train, there are three main routes available. In the south from Oslo (Norway) in direction of Göteborg (Sweden). In the middle from Trondheim (Norway) to Östersund (Sweden). And in the north from Narvik (Norway) via Abisko (Sweden) in direction of Lulea (Sweden) or Stockholm (Sweden).
The best route for you depends on your departure station in Norway. Please use the following booking links to find the best route, schedules and ticket prices. If you book early, you will find attractive discount fares.
Both countries offer very good night train connections to travel long-distance. The railway network in the North of Norway is not connected between Narvik and Fauske/Bodö. You have to take the bus for this part.