Based on two one week stays in Hennestrand in the late 90s and early 2000s and a weekend in Copenhagen in 2006.
Hennestrand is a small seaside resort on the North Sea coast, close to the German border. It has a long stretch of sandy beach with an extensive sand dune landscape behind the beach. The sand dunes are really very beautiful and, due to the dense vegetation, look like green hills from the distance. You can go on endless walks along the beach and return through the sand dunes. Behind the dunes, there are many colorful cabins that also look very pretty. We rented a cabin for the week over New Year and had a great time there. The climate was surprisingly mild and the weather was definitely good enough to go on long walks. The cabin was very cosy, complete with a fireplace and a little sauna. Hennestrand may be very crowded and touristy in the main season, but in the winter, it was very quiet and peaceful.
Copenhagen is a great city that combines the advantages of small and large towns: on the one hand, it's compact and easily navigable by bike, the streets are pretty and clean, the air is fresh and the general atmosphere stress-free and relaxed. On the other hand, it also has the cosmopolitan flair of a capital with many bars, restaurants and cultural events and a diverse mix of people including some alternative subculture. It's a walking and cycling city where you are mostly busy enjoying the atmosphere rather than visiting particular sights.
Beware that Copenhagen is not exactly cheap, so it's a good idea so have suggestions on reasonably priced hotels and restaurants. Useit Copenhagen is a great online resource for young low budget travellers. A very good place to stay is Absalon Hotel (check for cheap online specials through Sky Europe Hotels). Suggestions for eating out include a traditional Danish herring buffet at Faergekro and a Mediterranean buffet at RizRaz. Copenhagen also has great nightlife. If you are a student (or nostalgic about your student times), you could start the evening with a beer at Studenterhuset which often features life bands.
To get your bearings, walk up the Round Tower for a nice view of the city. A great feature of Copenhagen are its many canals, one of the most famous being Nyhavn. Once you have explored the city center, grab a bike (there are free city bikes all over town that work like supermarket trolleys: you insert a coin which you get back when you lock the bike again at one of the terminals), check out the outer districts by bike. Christiana in particular is worth a visit: it's a hippie commune that dates back to the 60s: lots of self-built wooden houses, cafes, art and the general alternative vibe that comes with a place like this. As you leave the so-called "free town" of Christiana, you will see a sign saying "You are now entering the European Union".