Start with amazing fish (haddock and cod), grass fed lamb and if you feel like it, order beef, because that's great too. (And from what I gather, all the livestock is organic - enough to make me buy organic meats). Beyond the great ingredients, the food was amazingly well prepared. In small, unassuming little places, we had amazing food. In Reykjavik, here is our favorite, if you make it to Akureyri, try this one. I don't think it is exaggerating too much to say that going to Iceland for the food is NOT a crazy plan.
A few things on most menus that we passed up were whale and puffin. I love to try a new meat and perhaps gain insight into another culture (water buffalo in Laos) but these two just couldn't excite me. Sorry if it disappoints , but it wasn't because I was opposed to eating smart whales or cute puffins - I tried to work up moral outrage and failed. It came back to the descriptions of the flavor - one description of whale meat as "like a steak" and another as "like liver" - what? Steak and liver taste nothing alike and I wasn't going to risk having a huge piece of internationally controversial liver in front of me. Puffin was described as "kinda gamey" and "really gamey" - not enough to convince me to give up "butter fried haddock" which I presume is what they eat in Valhalla (how's that for Norse flavor).
If we had made this trip 100, or even 60, years ago, we'd have eaten far more of the traditional Icelandic cuisine. For much of its history, Iceland was a poor nation, where nothing edible was wasted. Now, with a healthy economy, some traditional cuisine is still ubiquitous - Skyr and geysir bread (its name comes because it is cooked by steam in under ground ovens), but some less desirable options have been left behind except on a few holidays in the winter.
The photo i've attached is from a billboard advertising a cafeteria that serves traditional Icelandic plates (allegedly). The sheep head is not a reference to eating lamb, it is a reference to eating svið, preserved sheeps heads. Preserved in a gelatin of their own making, they can be "enjoyed" hot or cold. I never saw these tasty treats - and I went inside, read the dish of the day and spotted everything else offered - apparently these little treats are so unappetizing they can't put them out on display.
I also didn't see a whole lot of Hakarl - actually, I didn't see it anywhere - and I looked. This is the food mentioned in every guide book as a sort of open "double dog dare" to tourists. If you haven't clicked the link, it is usually described as "putrified shark". A shark too poisonous to eat fresh is buried in the ground for months until it is edible. The final product is said to have a strong ammonia smell and makes first timers dry heave. I survived durian, I think I would have tried it (easy to say now).