Piteå is situated 855 km north of Stockholm which is a considerable distance to drive. I would take the airplane any day.
There is no passenger airport in Piteå though so you'll have to fly to Kallax Airport in Luleå some 60-70 km north of piteå. From there you can take an airport taxi to Piteå for instance. It will cost you 165 SEK if you are willing to share with other people and agree to having the taxi drop you off at the taxi centre in Piteå. (from theer you can take an ordinary taxi if you want to, it will be cheaper than taking a taxi from Kallax directly to your final destination).
There are several flights a day from Stockholm Arlanda (ARN) airport to Luleå.
The best way is by airplane. The cheapest way is by bus (/train).
Stockholm 945 km,
Luleå 50 km,
Umeå 220 km.
New York 6332 km (3934 miles)
Bus, if you're going just outside the centre. But if you're just staying in Piteå, hey it's a small town, walk.