We didn't go quite in order as had been recommended by the city guide (mostly because our lunch destination wasn't near the first point of interest), nor did we see everything on the lists, but that didn't matter; we figured we'd see the rest another time.
In our search for historic sights, came across a park and monument to Jón Sigurðsson, a 19th century leader of the Icelandic independence movement. Across the street was the House of Parliament building, which was built in 1881, although Iceland itself has the oldest living Parliament; founded in 930 AD, it predates Great Britain's by 851 years, the United States Congress by 777 years, and Russia's Duma by 976 years.
At Fogetagardurinn Park, we saw the site of Reyljavik's first cemetery dating to the time of Iceland's settlement, and had been used for over 800 years; it is believed to contain the remains of 30 generations of Icelanders. We walked down Reykjavik's oldest street, Aðalstræti, excavations of which have revealed ruins dating back to the Viking eta. On Aðalstræti we were also able to see Reykjavik's oldest timber house.
After dinner, we took a short walk down by the sea to see the Sun Voyager. Just after 9:30 at night, the sun was still out and almost blindingly bright. It was too cold for us to take a long walk by the sea, but many others were out walking along the paths; it was quite a lovely night, a good one to see the sculpture.