If you are new to motor-homing, you’ll soon discover the not so much fun jobs such as dumping the grey water tanks, dumping the black water, filling up the water tank and maybe getting more propane.
Some campgrounds have all of these facilities, others have some but not others. The easiest thing to find is water hoses for filling the fresh water drinking tank and a gas exchange. Most gas stations have gas canister exchanges. Many have washing station with brushes attached to hoses for cleaning off the dirt on your car or camper. Water is typically available at campsites or gas stations.
Grey water which is the water from the sink or shower is typically mild stuff but it does contain grease and laundry detergent (i.e. phosphorous) so it can be dump just anywhere. I ran into one campground that had a black water dump but not a grey water. But a lot of the gas stations have drains appropiate for dumping gray water. Sometimes you can even park over a drain while filling up.
Black water is another matter. We used the toilet in the camper sparingly. Primarily for pee and used the campground toliets or swimming pool facilities for more substantial usage. As a result, our 17 liter little black water tank only needed to be dumped three times in 11 days and even then it was never rally full. (Three adults, using facilities outside the camper as much as possible).
As a result we were not too concerned about finding black water dumping places. We used them when we found a convenient one. It wasn’t until we had to drop off the camper that we had to search around for place to dump our crap. After searching the gas stations around the drop off point in vain, we stumbled on a great website that lists all of the dumping stations on the island:
http://www.ust.is/library/Skrar/Kort/kort-seyrulosun.html
Some of these location will be at campgrounds, some at gas stations and some like the one we found near the airport and the camper return was a stand along one down near some boat docks.