>>899355All forrests in DK, public and private are legally accessable during the day. Officially you aren't allowed in private forrests during the night but no one has ever stopped me.
In some public forrests you are officially allowed to sleep/camp wherever you want to (google "fri teltning"), in others only in designated camp spots - but again, no one has ever stopped me from setting up my tarp and sleeping wherever I wanted. I just keep away from large trails and out of sight of buildings and roads.
You are not allowed to light a fire in any other places than designated fireplaces - again I always light myself a small fire when I camp, especially if I'm on my own. As long as you're a bit stealthy about it, don't cut down living trees and don't leave behind trash and stuff I say do as you feel like.
As for places to go, I personally hike. Since nature is sadly sparse in this country a good start can be to follow marked trails as they tend to go through the bits of nature that we have. These are the most well-known and also longest that we have:
http://vandreruter.dk/vandreruterFor other information (like shelters and smaller trails) this site is a gold-mine
http://www.udinaturen.dk/ 
