Svolvær is in Lofoten.
There are express boats between Svolvær and Bodø Sunday-Friday, and Narvik. Both taking approximately 3½ hours. Further, you may take the Coastal Steamer, Hurtigruten, which arrives at 19:00 (southboud) or 21:00 (northbound) daily. Additionally, there is an airport in close proximity to the town.
Svolvær is really compact so the best thing you can do is walk.
There are many art galleries here.
It's not so easy to keep to the budget here. Narvesen kiosk is a good place to buy things such as drink. There's a bookshop in the town, art is being sold in many places and overpriced souvenirs are easy to find.
There's lots of different kinds of accommodation. Hotels are widespread, there's camping or you can get a rorbu (a traditional fisherman's cabin). Sleeping outside is possible as it is quiet, but the temperatures can be near freezing even in the summer and the cold wind from the sea doesn't help at all.