Iceland is not known for cheap food. Eating out costs about the same as any expensive Northern European city, and prices don't drop much outside Reykjavik. Self-catering from the Bonus or Kronan supermarket chains cuts daily costs significantly.
Traditional Icelandic food revolves around lamb, fish, and dairy. The lamb is genuinely excellent — sheep roam free in the highlands all summer, and the meat is lean and strongly flavored. Arctic char, cod, haddock, and salmon are all available and usually fresh. Skyr — a thick, mild dairy product somewhere between yogurt and soft cheese — is eaten at every meal and widely available in supermarkets.
The more unusual traditional foods are deliberately confrontational. Hákarl is fermented Greenlandic shark, buried and cured for months; the ammonia smell is overpowering. It's traditionally served at Þorrablót (the midwinter Viking feast held in January and February), accompanied by Brennivín, Iceland's caraway-flavored schnapps also known as "Black Death." Sheep's head (svið), blood sausage, and various offal dishes also appear at these events. All are worth trying at least once.
Reykjavik has a proper restaurant scene, with everything from New Nordic tasting menus to Lebanese fast food. The iconic Bæjarins Beztu pylsur hot dog stand near the old harbor has been serving lamb hot dogs with remoulade, raw onion, and crispy fried onion since 1937 — it's cheap, good, and very Icelandic. Kaffi chains are good for inexpensive soup-and-bread lunches.
Outside Reykjavik, options thin out quickly. Larger towns like Akureyri have a handful of decent restaurants. In the countryside, the petrol station is often also the café and the shop — the hot soup is usually fine.
Alcohol is expensive and sold only in state-run Vínbúðin stores, which close in the evenings and on Sundays. Bars in Reykjavik serve it without restriction, at predictably high prices.