Copenhagen for 4 days

You don't really need to decide in advance. There are some great day trips, but you might find yourself enjoying Copenhagen enough that you don't want to go anywhere else. So if I were you I would play it by ear. There's nothing that requires an advance reservation. Roskilde is an excellent option, but you can wake up one morning and just decide to jump on a train and go.

Check out the guidebooks and see what interests you - can easily spend all four days in Copenhagen but I enjoyed getting out of the city too.

These are easy train rides from Copenhagen:

Roskilde: Cathedral and Viking Ship Museum (which I found fascinating, but I’m into boats)

Castles (north of Copenhagen): Frederiksborg Castle (in Hillerød) and Kronborg (in Helsingør) are linked by a direct train and each is connected to Copenhagen as well, making a good triangular route.

You can easily spend four days in Copenhagen without feeling bored, but a day trip somewhere is not a bad idea either. It all depends a bit on what you feel like, are interested in and want to see and do.

There are also many options for daytrips to Sweden from Copenhagen. Lund or Ystad e.g. are good options.

We have just returned from a 5 day trip. Not my first time.

I agree with others. Copenhagen proper has a lot to offer and the day trips mentioned are easy and quick. Lay out your options and you can decide day by day based on weather, tiredness, spur of the moment interest, etc.

If you want something a bit farther, around 90 minutes by train, Odense is a charming town with Hans Christian Andersen heritage. Not any more of a must-see than the other options mentioned, but I hadn't realized until this trip that it is an option as a day trip.

We were very taken by the Danish Resistance Museum if WWII and its impacts is an interest to you. I mention that as it wasn't quite as obvious in some of the travel literature I reviewed as Copenhagen's art museums, castles, canal tours, etc.

We stayed 2 extra nights when SAS cancelled our return trip to Chicago. It worked out for us with some inconvenience and added expense (I have submitted forms for reimbursement, we will see....). I mention this as SAS has been cancelling on short notice with some regularity. More last summer I understand, but they are apparently still short of planes for any contingencies.

Enjoy!