Shopping with the dog
If you are going shopping, it is a good idea to leave the dog at home. Most shops do not allow dogs inside, unless it is a service dog.
On the other hand, you can usually take your dog with you to the market, which often takes place outdoors. Make sure that the dog do not disturb the other visitors, and keep a reasonable distance from places that serve food.
The dog at the restaurant and cafe
As a rule, the dog can not come inside restaurants, cafes and other places where food is served - unless the owner allows it. On the other hand, you are usually very welcome to bring your dog if there is outdoor service at the restaurant/cafe, and your dog can lie under the table so that it does not disturb other guests.
On the website hvrc.dk you can look up the city you are visiting - or the restaurant or cafe you want to visit - and here you can see if they allow dogs inside and if you have to book a table first.