You won't need that different clothes from what you're already used to. Maybe a medium thick wool layer and some mittens, a woolen hat in the beginning, and goretex outerwear. You can buy Janus clothes at Coop Obs Hypermarked once you're in town, and there are often price reductions, as well.
Welcome to Tromsø! I suspect you won't find the weather so different from what you're used to, at least not once the snow has melted.
Tjorven
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Tromsø in May can be absolutely wonderful or it can still be winter. If it is winter they will need wool. Norwegians get pretty good at finding wool at affordable prices. Wool does not need to be washed as often as cotton or synthetic fibers.
I buy a lot of clothes for my children at H&M. It is cheap and holds well. H&M has about the same prices in Norway as in London (which is what I can compare with).