Bit late to this but we've camped 3 times, once with children there.
For the ferry, consider getting a YHA membership, costs £18 but can get you 10% off. There may well be other discounts but agree it can work out expensive. Also think about taking your own food, maybe at least for the evening meals or if you are camping you may have a cool box anyway.
Campsites are often well run and have good facilities, many of them have a children's playground for example and often they have pools, kitchens. They are quite reasonable for the price but you often have to pay for showers, hot water is usually free, so we bathed the kids ina orblieb to save a bit of money or they came and had a shower with us.
For food, check out Netto's, kitchens are usually free and have good equipment.
Driving and public transport in DK is easy and also cycling, it's quick to get to most places. Admission can be steep as the price for children seems to be quite high from a young age but there's lots of free things to do.
My favourite camping places are-
Charlottenlund in Copenhagen but do book this or you won't get in www.campingcopenhagen.dk/gb/gbwel.html
Mons Klint www.campingmoensklint.dk/engelsk/default.asp
Faaborg (but can't find a link)
there's many others we have enjoyed though.
For Viking attractions, do go to Roskilde (can be done as a day trip from Copenhagen),
trelleborg near Slaglese and Jelling. There are lots of viking stuff and burial mounds dotted all over the place.
We also liked Egeskov castle on Funen, a day trip to Aero, and of course Legoland and Tivoli.