I would definately spend one or more nights outside of Reykjavik. It's a fantastically beautiful country - with prices to match.

Personally, I'd head to Snaefellsnes. It's a 2-3 hour drive from Reykjavik. You can catch a boat tour at Stykkisholmur. (Internet address below.) The tour takes you around the countless little islands in the bay and the birdlife there is something you will not have seen before. It's amazing. We were able to see an eagle there - what a wingspan, and of course endless puffins. The tour is I think was 2 hours.

You can then head to Budir on the south shore of the Snaefellsnes peninsula. We stayed at Hotel Budir (Internet address below) - fantastic restaurant - and the hotel organized for us a trip on the nearby Snaefellsjokull glacier, which Jules Verne made famous.

We had a fantastic time doing this (out last trip). But be warned: Iceland is more expensive than you can imagine...

http://www.seatours.is/

http://www.hotelbudir.is/