I've been to Reykjavik three times, for EvE Online fanfests. I think it's a terrific place and I really need to get back and drive (or ride!) around the whole island and not just the touristic Golden Circle route. I'll dig up whichever pics I have lying around to make a better contribution to this thread, but at the very least I should say that Hallgrimskirkja, the central cathedral in Reykjavik, should absolutely be on your list of things to see (you can see a little bit of it in the thumbnail on the first link in the original post here). It's so weird and unexpected architecturally and visually from several completely different angles... and with right lighting it can take your breath away. During my 2nd visit I accidentally found myself turning into Skólavörðustígur (the street that goes directly from the cathedral towards the harbour, slightly downhill), it was foggy and the sun was just starting to climb through the mist behind the structure, and it was an absolutely phantasmagoric vision framed by the utterly everyday street context. I'm purposefully not embedding any pictures of it so as not to spoil you from discovering it all by yourself, if only via Google.
Also, Reykjavik Roasters is an excellent 3rd wave-ish coffee place with Kaffitár being a great alternative (the latter boasts(-ed?) several World Barista champions but the former roast their own beans :)
If you fancy such things, you can find an EvE Online Players monument in the old harbour, and if you ever played yourself and were active at the right time, you may find your name on it using this handy locator service :)
Old harbour also hosts a kind of bazaar/fleamarket here, but please check the hours, I'm not sure it is open every day ; it also has decent local food options.