We did drive over, via DFDS ferries, it's a good idea to book early, plus we found that having a YHA membership got us a 10% discount, but check this out as it may have changed.
Driving is very easy, and to give you an idea of distance it takes about 4 hours to get from Esbjerg to Copenhagen using the motorway. There are some toll bridges and possibly ferries depending on where you go but all part of the charm.
Campsites on the whole have better facilities than British ones, as well as showers they will have kitchens/dining places and also a playpark and many have day rooms, swimming pools etc. This doesn't give them a haven feel, more like a youth hostel. You will often have to pay a nominal charge for showers, washing machine but most kitchens are free, however, one or two do charge. They are a nice way to meet other people but you can also cook by your tent if you want too. General camp stuff here
We have stayed in a number of places, but our favourites are-
charlottenlund which is in Copenhagen. Very very popular and I would recommend booking before you go. It's a bus ride from town, near a beach and an aquarium, plus a great location to get to places like Helsingborg, north of Copenhagen, Koge, Roskilde (excellent Viking museum) and Sweden.
Mons Klint is also nice with a pool and great play area, real fires, a few mozzies though for my DH (I was fine for some reason). It's quite a remote part of Denmark and gives an islandy feel without incurring extra ferry costs, which Aero and Bornholm do.
I like the area near Faaborg, and have stayed here and also here. It's a good base for the 'middle' Island, you can take a day trip to Aero, Egeskov castle, Odense etc
I'm not as keen on Jutland, which is the bit connected to Germany but that is where Legoland is and there are other attractions. The legoland campsite is very big with an airport fairly near by, personally I would prefer to stay elsewhere (near Silkeborg) and travel. Haven't found the perfect campsite there yet but there's a few other things to do- Aarhus has a great open air town musuem, plus there is also Tolland man, and Jelling is inspiring too. If you're into windsurfing, the area in the West is fab.
Denmark has a fair few islands- we have travelled over to the Aero twice, once when it was hot -fab and the other when wet- not so good. Bornholm takes a while to get to, we actually got there via Sweden but I believe you can go from Copenhagen but has a different feel to it.
HTH, feel free to ask if there's any more questions.