What rights do I have with a permit

What are you allowed to do with a Danish residence permit as an intern - and what are you not allowed to do?

You are only allowed to work in the internship position and for the internship host for which you have obtained permission.
If you are offered a new internship position, you must apply for a new permit.
This also applies if you are offered a new position, such as a feed manager, with the same employer. 

A Danish residence and work permit does not give you the right to work in other Schengen countries 

A residence and work permit allows you to stay in Denmark for the duration of the validity of your permit.
With a residence permit in Denmark, you have the right to free Danish language instruction.
However, you must be 18 years old and have registered your Danish address in the national register.

The portal borger.dk provides you with information, links, and in many cases, options for action regarding important topics such as:  NemID (electronic identification), CPR registration (civil registration), health card, tax matters, vacation arrangements, housing, danish language instruction, vehicle registration and driver's license