Helsingør is really cool. Kronborg Castle, a maritime museum, cheaper than Copenhagen, a big food market in a converted shipyard, and you can take a ferry to its sister city in Sweden that only takes a few minutes. Even if you don't stay there, it's worth a day trip. It's only an hour north of Copenhagen, but I was happy to have stayed there overnight and to have comfortably seen a bit of Sweden.
Æro Island is amazing. But, I only gave myself two nights there and I wish I would've had more time. It's not for everyone (super quiet, quaint).
Transportation in Denmark is ridiculous good. It's a very small, flat country and easy to get around via train and ferry. Jutland is even doable from Copenhagen in a few hours.