Iceland is a unique place and has so much to explore in the manner of nature as well as culture.
You can try to get a bit closer to the arctic and travel to the north of the country, another option is to take the so called Golden Tour to see Gullfoss, Geysir and national park of Thingvellir, a bath in one of the many hot springs all around the country, visit one of the many glaciers and take a adventours ride on the ice. Reykjavík, is one of the smallest capital cities in the world and it's easy to find your way around.
Just to name few things you can do in Reykjavik
Visit one of the many swimming pools in the city. Swimming in the warm geothermal heated water and relax in the hot pots. Belive us but this is a cultural place to visit, and Icelanders come there to socialice and exercise both in summer and winter time.
A wide variety of museums are found in Reykjavik you can surely find museums and art galleries that meet your interest.
The Church of Hallgrímur is a tourist attraction with a great view. The church is 74 metres (244 ft) high and is the largest church in Iceland.
Walk around the Lake Tjörnin in the centre of Reykjavik and explore the birdlife. Just sitting by the pod and enjoying the moment and listen to the city life.
If you want to have a good view over the city you should explore the Pearl (Perlan) on the top of a area called Öskjuhlið. The Pearls sits on three big tanks that hold Reykjaviks hot water after it is pumped from geothermal wells. In the Pearl there is a Viking museum, cafe, restaurant that spins around and overlooks the city.
Few things you can do outside Reykjavik:
The Sagas - Iceland´s most important cultural heritage
Looking for places of interest in West Iceland? The Settlement Centre at Borgarnes is a unique place to visit. The Icelandic Settlement Centre at Borgarnes tells the Saga of The Settlement of Iceland and Iceland´s most famous viking and first poet Egill Skallagrimsson.
The Settlement Exhibition provides an insight into the settlement of Iceland. It tells how the land was discovered, how the Viking sailors conquered the open ocean and why they left their homelands in Norway. The Egils Saga Exhibition profiles one of the most colourful of all the saga heroes, Egil Skalla-Grimsson was the greatest poet of Iceland in heathen times. He was also a Viking and raided and pillaged widely throughout northern Europe.
It is not a museum but rather an instalation. Multi-media and theatrical techniques are employed to help the visitor experience first hand the trepidation and excitement of setting off over the open ocean for lands unknown.