This is a lovely campsite if you need one in Copenhagen but I advise you to book.
How much travelling do you want to do? I love Skagen in Denmark but it won't be en route if you need to get to Sweden quickly. If you did decide to head that way for a few days you could get the ferry across from Arhus or Ebeltoft which would save a bit of driving (good fun too if your vehicle is small enough for the fast ferry).
Other places I like in Denmark are Ebeltoft, Arhus, Aero, Faaborg (good for a day trip over to Aero), and the area north of Copenhagen.
We have camped in Sweden but didn't enjoy it as much. We made our way over to Simrishamn and also had a few days in Ystad. An alternative could be to get over to Bornholm, either from Ystad and I think a ferry also goes from Copenhagen?
There are plenty of Lidls etc to get a range of cheaper food, although food prices are now quite comparable. A lot of Danish campsites have kitchens you can use (generally free) which saves on your gas etc, but it helps to have a box/bucket to get your cooking things to the kitchen. Are you taking children? Campsites are well suited for children. Also, on the ferry you can take a dvd player up (take a plug adapter) which helps, plus we usually have a meal at Morrisons in Harwich, and then take a picnic up to our cabin. Take extra water up too!
If you want more details on Denmark, let me know if you are thinking of particular areas. Or details on what you would be interested in seeing in Denmark- if you like design museums, Viking stuff etc.
We're heading to Scandinavia in a campervan for 2.5 weeks in July - sailing from Harwich to Esbjerg and then driving across Denmark and spending some time with friends in Copenhagen before crossing 'The Bridge' and spending some time at the beaches/lakes of Southern Sweden.
Never been to Scandinavia before - any tips? What to do? Where to camp?