Favorite thing: If you like art or would like to see some local handcraft be sure to visit Skólavörðustígur. It is a street that goes down from Hallgrímskirkja church to Laugavegur main shopping street. It hosts several galleries and handcraft shops. If the weather is ok you could go up to the tower of Hallgrímskirkja (approx 70m) and get a great view of the city and surroundings.
Favorite thing: Visit the Hallgrimskirka, the huge church on top of the hill that dominates the skyline of the city. The view from the top of the tower is outstanding (beware of the earsplitting bells though, if you're up in the tower!)
Favorite thing: Hallgrimskirkja church Construction for this church was started in the late 1940's and was completed in 1974. It is said that the exterior of it resembles basaltic lava. It is Reykjavik's tallest structure and can be seen from over 20 km away. . There is a great view of the city from its 75 m tower.
Favorite thing: While you're at Hallgrimskirkja, the huge church that dominates the city skyline, be sure to see the statue of Leif Ericson in front of the church.
Favorite thing: See the Leif Eiricsson Monument located near Hallgrimskirkja Church. This statue was presented to Iceland by the USA in 1930.