With the caveat being that I haven’t been for a few years, I remember the following being good.
EDIT! I just noticed you said family… some of the spots might not be super family friendly (especially at night) so worth Googling first.
Vesterbro (area) - former industrial area now gentrified with plenty of good food/drink. Think London’s Hackney Wick or Newcastle’s Ouseburn.
I loved Warpigs which is a Mikkeller craft beer bar with bbq.
Mother for great pizza.
Kihosk was a cool neighbourhood off-license style shop with loads of seating outside. Ideal place to buy a nice beer and sit outside on a Summer evening.
Bæst - Amazing pizza in a more restaurant style environment than the above. Pretty sure it usually sits high on ‘world’s best’ lists.
Next Door Cafe in the old Latin quarter was a nice little cafe known for amazing homemade blueberry pancakes.
Unhelpfully I haven’t got any good coffee recommendations other than Coffee Collective which I think is a small chain but they knew what they were doing and the coffee was very good.
Honorable mention for Meyers Bageri which I think is also a chain but they had a branch near to the place I was staying in Østerbro and the croissants were so good I think I went every day I was there.
Obviously you’ve got some very high end foodie spots like Noma and Alchemist but it’s not really my scene so I can’t comment.
Unless you’re into touristy spots I’d maybe not spend much time in/arount Tivoli and the Central Station.