There are a lot of possible day trips from Reykjavik to explore the surrounding area. If you only have the chance to do one I would suggest the Golden Circle, the tour will pick you up from your hotel and return you to it at the end of the day. During the day you will see geysir - the water spout that others are named after. Whilst geysir doesn't erupt any more strokkur next to it goes off every few minutes and is an impressive site. It also stops at Gulfoss which is a beautiful waterfall, and Þingvellir national park - a lovely lake where two tectonic plates meet.
I did this tour in 2009 and really enjoyed it, and from the website it doesn't look like things have changed massively.
Another thing you might be able to fit in on the way to/from the airport is the blue lagoon - some companies offer a single ticket from Reykjavik to the blue lagoon to the airport at keflavik. It is a relaxing geo-thermally heated pool, but is fairly pricey.