Copenhagen is a wonderful city and it is perfectly designed for exploring with kids. From parks, to attractions to the attitude of the locals, we had an amazing experience, even with three little ones in tow.
Denmark is child friendly, and I saw far more children around than you would in London at this time of year. Everywhere we went was pushchair friendly, and there were always lifts or ramps making it accessible. As their schools start much later than ours a child is generally considered to be over the age of six, meaning our three were free to enter all the attractions we visited.
Staying outside the city centre had some definite advantages for travelling with young children, one of them being there was a beach only a short walk from our house. Whilst we didn't spend much time there, and it wasn't really the weather for sandcastles and swimming, it was lovely to balance out a bustling city with a peaceful seafront. The boys had the freedom to run and explore and it was the ideal place to go after a couple of hours cooped up on a plane.
At the tourist office, you can purchase a Copenhagen card, which is valid for a period of time (24,48 or 72 hours) and gives you unlimited free travel and entry into a number of the city's main attractions. There is a handy calculator on the website that helps you work out whether it is worth it, and we found it definitely was as it covered most of the places we wanted to visit and we could use whichever transport we chose (it even includes transport to and from the airport)