1. Blue Lagoon and Reykjanes peninsula

Blue Lagoon is maybe the most famous spot in Iceland. It takes about 40 minutes to drive to there from Reykjavik. Head towards Keflavik in the Reykjanes-peninsula, but turn left some 10 km before entering Keflavik and pretty soon you are at the Blue Lagoon.
Before entering the lagoon, it can also be nice driving around Reykjanes. There are many 'beaches' (maybe not to sun-bathe in but nice to walk around and pick up some shells and play), lighthouses and also the 'Leif the Lucky Bridge' which spans the Alfagja rift valley (60 feet wide and 20 feet deep) near Grindavik that marks the boundary of the Eurasian and North American continental tectonic plates. It was built in 2002 and named in honor of Icelandic explorer Leif Eriksson who travelled from Europe to America 500 years before Columbus.