I spent a few months there of my 6 month deployment. I was there during the winter months so I am sure things are a little different. There is the obligatory tour of the Blue Lagoon. Try to get to Reykyavik(sp?). It is a great city, and the drive isn't too long, maybe 45 minutes or so. I rented a car on the base, but this was almost three years ago, so I don't remember the name of the place. Try checking out the Liberty program on the base, it is an MWR run program that caters to single military that are in Kef. PCS or TDY. They plan outings and such. I really enjoyed my time there, but that might have been because the previous 4 months of that deployment was spent on the Navy base in Puerto Rico. Thank god they closed that hole (no offense to anyone who liked that base, just my opinion).
Have fun.