When you're in Piteå you absolutely SHOULD rent a car or find a friend to drive you to Storforsen, which is a nature reserve and a large white water rapid in the Pite river. The rapid itself is situated in Älvsbyn municipality so I've put my tips on it there, but here you can see a travelogue of the place:
Sweden was until the mid of the last century a very religious country and the Lutheran church wasn't separated from the State until the 1990's. Before 1900 all people went to church and up until the 1860's it was a crime not to go. Many people had long distances to their church, especially up in the north and church villages was brought up where people could live while attending church.
The most famous church-village is Gammelstadens Kyrkby in Luleå. The village is even on the UNESCO world heritage list. The church village in Öjebyn is much smaller but well worth a visit. Especially if you go to the tourist information office in Piteå first and get a map with a löegend to what all the houses were, such as prison or butcher's house.
Öjebyn, just some 10km to the west of the city (almost a part of the city actually) and it's well worth a visit. It was here that the OLD Piteå lay and it has a history spanning several hundreds years back at least to the 14th century. There is a beautiful church here that you should go visit.
This is one of two squares in Sweden from the 17 th century where the original shape is preserved (the other one is Stora torg in Uppsala). Around the square there are several early 19th century buildings.
The first church was finished in year 1425. In 1750 it was rebuilt. Around the church there is 'Kyrkstaden', overnight cabins which were used by people who had a long way to the church.
Piteå Dansar och Ler (Litt. Piteå Dances and Smiles)
A festival with alot of the best swedish artists.