Are you flying in on an overnight flight from the US? If so, I would not get a car as it is not wise to drive very far at all, roughly an hour, on arrival day. I would head into Reykjavik upon arrival. There is the Flybus which is about $38 per person and makes a stop at the central bus terminal and you change buses to a smaller one that will take you to various bus stops in the city. A taxi though would be significantly easier for you with 2 little ones, and those run about $150.
So get to Reykjavik and check in at your hotel and leave your bags, and go wander the city. There is a lot to see and do there. You could hit one of the local pools as well.
In the morning you could head to the Blue Lagoon and enjoy your time there before heading back to KEF. Or you could rent a car for the half day and do the GC. A bit rushed, but doable.
If you are not dealing with an overnight flight, then renting a car upon arrival and doing the GC would be fine.