Bringing your kids to Copenhagen you can find a lot of entertainment. Outdoor there are playgrounds in all parks, you can go to the Round Tower, which is great for kids, with its spiral walk way. In the summer it is free of charge to go swimming at the harbour pools, or you can go to the lowwater beach at Amager Strand.
There are two amusements parks worth mentioning. Tivoli in the center of town, can be a real treat, but it can also be a day of endless lines, and you feel like you are using millions on worthless shit. If you are on a budget, bring a picnic basket. Check the website for opening day and hours
The cheaper alternative is Bakken, north of Copenhagen, placed in the Dyrehaven forest. (C train to Klampenborg). There is no entrance fee, and you can take a walk in the forest, where you find a lot of deers. But the eating is real shitty fastfood, so if you are a little bit healthy: Bring a picnic basket. Open only in summer, and website is in Danish and Swedish
The Zoo is on the top of the hill, Valby Bakke, and is well kept with the new elephant house, and the polarbears also have a new area for themselves. And soon their will be Pandas !!! The Zoo is a bit expensive, but it is entertanment for hours. You can borrow a small four whell trolley for the small kids and your bags.
For the bigger kids there is the Experimentarium in Hellerup, north of the center of Copenhagen. It is a science hall and the kids can try all sort of stuff, and even you as an adult can learn something 😉 .