The Danish climate can go from cold to hot and then back another time in a few hours, so coating are your best bet. Bring large numbers of sweaters and at least one water-resistant jacket. Most Danes own a complete set of rain clothes, a kind of rainproof jogging outfit. For winter, you will need temperate scarves and a warm coat - a short one is more sensible, since a lot of people get around on their bikes all winter long.
Even for summer, plan on bringing a a small number of sweaters and a solid jacket - leather is ideal. Danish summers are often rainy and cold.
You will be doing a lot of walking, so bring easy shoes, water-resistant if possible. Danes have a tendency to dress in subdued colors - brown, navy, grays - that match the colors of nature in Denmark.
For women, long skirts are more sensible than short, even for summer, since skirts that are knee-length or shorter are too helpful when riding a bicycle. Danish women wear slight makeup and plain hairstyles, so there's no cause to drag along a suitcase of fancy products. High heels are a terrifying on paved streets, so if you must wear them, bring down a pair of flats to wear until you reach your destination.
You do not need much fancy clothes, unless you're a real nightlife enthusiast. And unless you have a job in a bank, there is no reason to bring more than one business suit. Jeans without rips or holes is perfect worn almost everywhere.
You can for all time buy the clothes you need when you reach your destination in Denmark, because prices are at least double and sometimes triple those elsewhere, once 25% sales tax is factored in.