What to see or visit outside Copenhagen really depends on what you like. Some of the suggested places and sights are excellent, and at http://www.visitdenmark.com/ you may get some other ideas and inspiration. Strangely the west coast from southern Jutland to Skagen at the northernmost tip is not listed as an "attraction" although it attracts thousands of tourists (mostly from Denmark, Germany, Norway and Sweden) every year.

Regards, Eric, from Aalborg in northern Jutland, Denmark