The National museum of Iceland is your best choice if you want to sneak a peak into the history of Icelanders. It will take you through the times of early viking ages through the christian medieval times to the technical history of the 20th century. If you are ready to spend here a (rainy) day, you can really get an whole overview of the icelandic past.
How to Find National Museum of Iceland:
Suðurgata 41, 101 Reykjavík. The closest bus stations are the Háskóli Íslands (1, 3, 6, 14) and the Þjóðminjasafn bus station (13)
GPS coordinates of National Museum of Iceland: 64.141632, -21.948658 Take me there!