Eat skyr! Buy sweaters and sagas and a Sigur Rós CD.
There is an English newspaper for Reykjavík that may help on city life:
How long will you be there? My best advice is first to go soak in the Blue Lagoon and contemplate the meaning of life. There is a lot of curious landscape right there, not 30 mins from the airport. There is hydrothermal plant right there that is interesting to visit.
If you have more time then the thing to do is head towards Thingvellir. En route there is a cute little volcano named Kerith and a rather majestic waterfall, Gullfoss. All of this can be easily be done in a day trip from Reykjavík and will provide a good intro.
For northern lights you may want to get out of the city, in which case just driving east on the ring road, just over the ridge will cut down a lot on the light pollution. You can get the northern light forecast here:
You can also look up the latest weather, quakes and volcano eruptions here:
If you want more adventure I can advise on lava tubes and such.