Being in a country like Iceland, where the fishing industry is one of the most developed, you need to try to go sea angling. It is more than a nice trip, it is an experience that you will always have in your soul. I have been two times doing this, and I have never been disappointed.
There are two main companies operating the basic tour, but there are some that are doing longer trips and also private ones. My trip was around 4 hours in total. In the begining, the boat will take you away from the shore are usually they have a radar to follow the banks of fish, Then, you will get a short training about techniques and safety during fishing. They will stay for around 2 hours in the sea for you to catch something, and then they will start moving to the shore. On the way back to the harbour, the staff is cleaning the fish and they will make barbeque on board. I think it is the best fish that you can eat and maybe you will never have again the opportunity to eat such a fresh arctic fish.
I think it is one of the most interesting activities that you can have in a country like Iceland, especially because you are doing something, enjoy being away from the city in such a pretty enviroment and if you are lucky enough, maybe you see also the puffins. The staff was absolutely nice, they were never very close to you. Basicly, they were watching everybody to make sure that you do not have trouble and give you space to enjoy.