There are the usual Danish versions of all the usual supermarkets. Food is expensive due to the 25% MOMs (equivalent to VAT) on all items. Funny how a socialist country has a tax on the most basic essential of all, don't you think? Like the cars, apparently without the tax Danish supermarkets are the cheapest in Europe.
The main staple is Rye Bread, which I'm quite a fan of. Healthy stuff too, and if you want to eat it like a real Dane, then you'll have a "smørrebrød" plate with a massive choice of meat, fish and chocolate toppings to put on it.
Denmark is also proud of its fact of being a nation with more pigs than people. So pork products are very popular, although since pigs were historically a source of great export earnings, the Danish pork cuisine is based around what you can do with the bits that are left after you've sold the best stuff to Britain and Germany. Hence do NOT expect to find good quality bacon anywhere in Denmark, instead we find such delicacies as flæskesteg or "fried flesh" and rullepølse or "rolled sausage" - the first one is fairly self-explanatory, and the second is kind of off-cuts compressed into a square which can then be cooked like a joint of meat - despite how it sounds both things taste nice.
Dairy products are also another main staple of the national diet and I would have to say that the quality and choice is impressive. I have never seen a nation where the adults drink as much milk as here, and I suspect butter consumption per person must be about 3 or 4 times higher than in the UK.
Another famous item is Danish pastries. Confusingly enough they are called Wienerbrød here - or "Vienna Bread" as the sugary cakes were actually the concoctions of Viennese immigrant bakers to Denmark a few hundred years ago. Your local bakery, or even the supermarkets will stock a fine selection for you to choose from.
The Danes are big, big coffee drinkers, even more so than Italy and France, apparently, and no home is complete without a coffee machine and a trendy coffee flask for everyone to help themselves to a cup from. Tea is available in the shops, though it is packaged and priced as an expensive luxury for occasional use only.
As for a traditional takeaway, I recommend the hotdog stands and huts you find in all towns and cities. The hotdogs are much better quality than anything you can find in the UK. not very filling though, so you might need two or three washed down with a beer before you proceed on your way.
Now to a subject of great interest to me, most Danes and maybe to you too - beer. For years, Denmark has been dominated by Carlsberg/Tuborg and in my opinion these are mass-produced, average tasting lagers at best. However, last few years there has been a huge surge in micro-breweries, many of whom are experimenting with darker beers Brits are more likely to recognise. An excellent improvement, and the supermarkets are also stocking some of the British beers I miss so much, like London Pride. So availability of alcoholic beverages is something you won't miss out on.