We will be driving to and from Legoland from Copenhagen and would like recommendations on things to see, side trips to take, etc. We are a family with young kids and would like recommendations that will appeal to the adults in the group and others than will appeal to kids.
In Copenhagen, if the weather is nice, hanging out at one of the gardens is super cool. Kongens Have is one of the great ones to hang out in. The kids can play around with a ball, bring a frisbee, there is even a kids theater that does daily shows in the garden called Marionetteatret i Kongens Have. There is also a great lunch place in the garden. The name is Restaurant Orangeriet; for lunch they serve classic Danish fare and is reasonably priced. Great outdoor seating and a great place for families. When the kids (and the adults) are ready for an icecream go to Kafferiet, a small coffee bar in one of the quaint buildings that surround the garden. They have great ice cream from the artisinal ice cream called Hansen. You won't regret it. If you are looking for coffee though, the best place nearby would be Café Det Vide Hus. Great, great coffee. Some of the best in town. They also do a really good breakfast of skyr (a thousand year old Nordic low-fat dairy product) with seasonal berries, a soft-boiled egg, and slivers of avocado on rye bread. Just great. If you are in Copenhagen around mid-July onwards, King's garden also has four ancient mulberry trees producing, guaranteed, the best berries you will ever taste. See if you can find them.
Besides that, I would reccommend you to go to Papirøen. Out there you will find two things: one is the Experimentarium City, which is a sort of science playground. The kids will spend hours there, easy, and it is not to bad for adults either. Right next door you will find Copenhagen Street Food, a food hall with a dozen or so vendors selling chicken rotisserie and all sorts of other grab and go food. You can sit by the water and pick and choose from the stalls and there is lots of space for the kids to fool around.
Another fun place in town that works for all ages is Torvehallerne. It is a sort of upscale food market with great coffee, lots of food stalls with (for Copenhagen) reasonably priced small food offerings. There is also lots of fresh food on display. My kids love to hang out here and gaze at all of the colors and shapes. Right nearby, for a fun little trip, there is also the Arbejdermuseet (the Worker's Museum). It is fun to see examples of apartments from the old days of working class Denmark. Lots of fun small details.
While we are talking about old days, a fun place to visit on a sunny day for the whole family is Frilandsmuseet. It is an open air museum that tells a rather romantic story of Denmark with live animals, old buildings, green spaces. What's not to enjoy.
Hope this helps a bit! Have a lovely trip.
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The Egeskov Castle is a great place for both adults and kids if you have not been there before.
The castle is the best preserved renaissance water caslte in Europe and they have some very scenic gardens there too as well as a vintage automobile museum.
The place is quite family orientated too with some great stuff for kids so I don't think they will get bored there.
Check out their website: http://www.egeskov.dk/en
Make a stop in Odense (Home of H.C. Andersen). Here you find museum and an many nice places. When you drive over the second bridge to Zealand you can stop in Roskilde at the Viking museum - where you can also see many other attractions.