The State Archives
One of the main Danish sources for family search is The State Archives (Statens Arkiver). Many archives have been digitized including church records and censuses. Some have been made searchable.
The site is www.sa.dk with a reduced English version www.sa.dk/content/us.
The oldest living persons in the family
Be sure to interview the oldest living persons in the family, they often have interesting stories and myths to be explored, it will all be lost if not written down.
If you are interested in using the Danish sources you can take a look at some material that I have produced. For the time being it is in Danish. You can use Google Translate to get an idea of the meaning in English: Material-PDF