We have been a few times, with and without children.
The ferry is ok, quite expensive but we were member os the YHA so got 10% discount. Book as early as you can for the cheapest rates. We didn't bother with the evening meals, had a hot lunch at Morrisons in Harwich and then bought picnic stuff to take on. We did do the brekkies which are reasonable value if you eat lots. How old are your children? There is an entertainer on board but apart from that it can get quite boring. We took the car dvd player up to our cabin with a plug converter which helped.
Campsites are of a good standard, nearly all have some sort of a children's playground and a kitchen equipped with hobs etc, then there are various add ons- swimming pools etc. It depends on what you what, we prefer slightly smaller sites over the caravan type ones. You need a card which can be bought on arrival for about a tenner, more info here.
There's loads to see and do, especially of a historical nature, legoland of course, we spent 4 days in Copenhagen and got a 3 day pass thing last time which proved good value, got into the zoo, various design museums, Tivoli, a canal tour, aquarium and this amazing place . We also enjoyed the Viking ship at Roskilde, Arhus for Den Gamle By (open air museum, lots of little house etc to visit) and the Moesgard museum with the Grauballe Man, I would also recommend a trip to Skagen, so dramatic. On Funen, there is Egeskov slot and the ladby ship museum. Trapholt is also a great place.
Nearly all of the campsites we stayed at I would recommend, and we have returned to a few. My absolute favourite is the nearest to Copenhagen Charlottenlund fort. Ideal for visiting the capital, you can get a bus from outside the site into town. It has such a great international buzz, the downside is that it doesn't have a playground but it does have a swimming beach nearby and also the aquarium is near too. If you want to go to this site you will need to book in advance.
I found food to be a similar price to that of the UK, wine/beer is also about the same. There are many Lidl and similar dotted around. If you need nappies take them, it cost us about £15 for a smallish pack. Going to museums can be costly but there are often family passes. If you take any smaller ferries across islands, booking in advance even by a few days gives cheaper tickets, I managed to book a ferry from Arhus-Zeeland using the oldest computer in Denmark at a campsite and google translate for half the turn up price. The bridges are exciting to cross too but there is a toll, you can use a credit card.
I would consider doing a 2 or 3 site holiday. If I had 14 days, my choice would be up to Skagen, then get the hydrofoil from Arhus/Ebeltoft to Zeeland and going to Copenhagen, then coming back via Funen and the bridges to Jutland for legoland.
HTH, let me know if there's anything else you'd like to know.