We took advantage of the Copenhagen card for transportation around northern Zealand and admissions to various sights. Our remote trips included about a half day to Roskilde (to see the Viking Ship Museum and Cathedral), and a full day going to Helsingor (site of Shakespeare's Elsinor Castle) after a look around the town and visiting the castle, we took a "private train line" up to Hornbaek beach (private line also included on the Copenhagen card).
The trip we didn't take, but will next time, is to Hillerod to see Fredericksborg (the track was being maintained so it was a very inconvenient trip to see it when we were there).
In town, I'd certainly recommend Rosenborg Castle. It's a nice small-scale castle with the crown jewels in the basement. Amelienborg depends on what you like, since it's the current palace rather than something historic.
The Glyptotech and National Museum downtown are both worth a visit. If you like modern art, there is a stop on the way to Helsingor to visit.
Tivoli is worth seeing. My gripe is you pay admission then pay for anything else inside the park (but admission is included with the Copenhagen card).
We also enjoyed the boat tour from Nyhavn.
All those barely scratch the surface of what's available in and around Copenhagen.
The logical Swedish connection from Copenhagen is to ride the ferry across at Helsingor, then train back into Copenhagen through Malmo. I didn't think there was anything special there so we skipped it (we were going to Stockholm later as part of our trip).