Reykjavik

They say Iceland is the country of four seasons at the same day. Unfortunately when I was there it was all-day rainy, windy, umbrella-breaking WINTER! Come to think of it, many other places of the world (i.e. the Acores) are supposed to have 4 seasons in 1 day, albeit on the warmer, shinier side.
However, Reykjavik is the most northern capital of the world and as such deserves a stop, even if you can’t wait to get lost in the wilderness of the Icelandish, northern-European Far West!
2 out of 3 Icelanders live here, 130.000 in central Reykjavik, almost double as much in the greater capital region (total population of the country: circa 372.000). The city centre is very compact and you can see most, if not all places of interest in just one day. This will leave you also time for cinnamon rolls in Braud and co, a must of your stay in Iceland.
As for the rest, Harpa, the concert hall and conference centre should be your first stop. While there, don’t miss out on the How to become Icelandic in 60 minutes. Absolutely amazing!
The icelandishly austere Cathedral (Domkirkjan) has fantastic views over Iceland and Reykjavik. Be there on the hour to listen to the church clock.
Another must-see seems to be the Parliament House. I never understood why...
For the rest, Hafnarstraeti is the central road you are going to find yourself returning for almost everything while in Reykjavik: eat, drink, buy, see and be seen.

I cannot stress how amazing the spectacle Fly over Iceland is. I wish it lasted 3 hours, and not just half an hour. Not far from there, the Whales of Iceland is also very interesting, although sad. Interesting food at the harbour in the vicinity.
There is interesting shopping in the southern end of Hafnerstrati and in Harpa.