4 days in Copenhagen

You have plenty of time to do a little research on your own.
Copenhagen is a wonderful place.
My suggestions are to get your hands on the RS "Scandinavian Guide" or the RS "Copenhagen & Best of Denmark" snapshot books, the more current the better, watch the RS "Copenhagen" and "Beyond Copenhagen" segments of RS Europe (you can watch them right here on this site), and begin to form an idea of what seems to be of interest.

Here's a good start: https://www.ricksteves.com/europe/denmark/copenhagen

What are you interested in doing/seeing?

June/July should be good bike riding temperatures. http://denmark.dk/en/green-living/bicycle-culture/
http://www.visitdenmark.com/copenhagen/activities/copenhagen-two-wheels

You don’t need to bring a bike to Copenhagen to see the city on two
wheels. There are loads of bike rental shops dotted around the city.
Expect to pay around 75kr for a day’s rental and 350kr for a week.

For an even cheaper alternative, you can borrow a city bike to tour
the centre of the Copenhagen. They only cost 25kr per hour and they
come with a built in GPS! You can borrow and give back bikes and
stands all over Copenhagen city centre.

When traveling to any new city, we research the town on Wikipedia.
We stayed in Hotel Absalon in Copenhagen, 1 block from the train station and the Tivoli. Their provided breakfast was absolutely the finest breakfast we have ever experienced. The hotel has now been completely renovated and is quite comfortable. And they are so easy to get to.

I was in Copenhagen a couple of weeks ago for my first visit.
It's quite expensive!
The food and coffee are great.
My best days were two trips by train out of the city.
The first to Roskilde, where there is a beautiful Cathedral, and a really great Viking Ship museum.
The town is small and very picturesque.
The other day was a trip to to Helsingor, for the castle and the Danish Maritime museum, both excellent; and again, small town with character.
Quite different in winter than it will be for you in summer.

I spent 6 nights and 6 days in Copenhagen.
1. Recommend that you purchase the Copenhagen Card. Check it out and it tells you what it is available. It provides you with entry into numerous sites in Copenhagen, Kornberg Castle (Kornberg which about 1 hour away by train); Louisiana Museum (can see going or coming from Kornberg); Viking Museum in Roskilde (I believe); a boat tour around the harbor; and all trains/buses both in Copenhagen to Roskilde and to Kornberg.
2. Kornberg and the Louisiana make a great day trip out of Copenhagen.
3. Trivoli was fun but I wasn't much into riding the rides. But it was great to see all ages (babies to seniors) milling around.
4. Plan to walk, walk, walk
5. Have lunch in Nyhavn or dinner. It is quite lit-up at night.

Have Fun! The Danes are nice