Just a short train journey from the centre of Copenhagen takes you to Sorgenfri, the closest station to the Frilands Museum. It is about a ten to fifteen minute walk from there, alternatively you can take a bus from Lyngby Station. You can check your journey plans here.
This is an amazing open air museum with historic houses (which you can go into), windmills (the largest is open for visitors over the summer holidays), farmland, orchards and woodlands for explore. I would recommend bringing your own picnic (there are lockers by the ticket office) and, if there are any left, you can rent a trolley to pull smaller kids around as well as your lunch. The restaurant on site is not run by the museum and the food is not great.
Over the summer they have a folk dressed in historic costume demonstrating a number of crafts and skills such as log splitting, carding wool and making strong yarn, amongst other activities. The people working there are happy to speak in English and are very knowledgable about the history of the houses and the different traditional skills.
There are also some small theatre shows to watch and variety of other activities for the remainder of the summer holidays. There are pony rides and also the horse and cart ride around the museum.
There is an entry fee to the museum for adults but kids are free. You can also buy a very inexpensive year ticket which is well worth it if you plan to come again. You will also need MobilePay or cash for the ponies and cart rides.
You can easily spend the whole day here and it is a relatively inexpensive day out for families.
Here is the link to the website, this is the Danish version as there is a lot more here than on the English pages.