I'm not Danish, but I second Århus and the Moesgaard Museum (especially for Grauballe Man!).
In Århus also is Den Gamle By (apologies to our Danish friends if I have misspelled it!); well worth a leisurely stroll around - it's an open-air town museum (The Old Town), various jolly nice half-timbered buildings and so forth. If I recall correctly C15th through C19th.