I would suggest Luebeck.
If you have 5-6 days available:
1) possible stop in Lübeck -> ferry + train to Vordingborg in Denmark -> bus to rural Møn Island and a B&B there
2) Via north German coast to towns like Lübeck, possibly Rostock or Stralsund; then Sassnitz - Rønne ferry. From Rønne town bus to Allinge-Sandvig. Exploring rural Bornholm island, B&B or hotel in Allinge-Sandvig.
Sights: Hammershus Castle, Hammeren peninsula, Gudhjem village; Christiansø islet, Allinge røgeri, Svaneke, Østerlars church etc. etc. Rented bike in Allinge recommended, also, there´s a lot of buses available.
Connection to Copenhagen 3-4 hours.
Public transport in denmark: www.rejseplanen.dk
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