Hveragerði is a short trip away from Keflavik International Airport. Once there, if you're going to want to participate in the wedding festivities, you'll need to continue through town to the Frost & Fire Hotel at the address below. This is where the fun begins and you definitely don't want to miss it.
Frost & Fire Hotel
Hverhamar, 810 Hveragerði, Iceland
This is likely the preferred method for getting around as Iceland's road system is great. And if you're going to be driving, the process is fairly straightforward: get a car, a GPS (or a map if you're one of those people), some snacks, and hit the open road! Just head for Hveragerði and you'll find it in no time.
Driving in Iceland is more or less like driving in the United States and can be accomplished with a US driver's license. Of course, it lies just south of the Arctic Circle and is, in fact, an island. As such, it's weather can be somewhat unpredictable. Also, volcanoes.
The Icelandic Transport Authority provides some standard info for getting around and you can look right here to bone up on Icelandic road signs. But if you're really into it, here's a video about the joys and perils of Icelandic driving. Or here's another if you're more into elves.
Public transit takes two primary forms in Iceland: taxi and bus. The likes of Uber or Lyft have not arrived on the island and likely won't be there by July '18. Furthermore, getting out to the Frost & Fire Hotel via this method is a little tricky but can be done.
From their website: "To get to Reykjavík from Keflavík airport it is best to take the Flybus. If you are stopping in Reykjavík at all the flybus offers a ride to the hotels in Reykjavík. To get to here by bus you´ll have to go to Mjódd bus station in Reykjavík and then take bus nr 51 or 52 to Hveragerði. The bus stops at the Shell gas station in Hveragerði and from there you could get a taxi to the hotel or walk."
It's a little more circuitous than renting a car but you'll get there eventually. Also remember that other people are (ideally) coming to this wedding. So one person's car hire is another person's public transit! Ask around or discuss with fellow attendees here.