First, you need to understand that all of Scandinavia is very expensive. So you need to give us aspecific amount that you wil consider reasonable.

As for what to see and do - why did you decide to go there? You must have had some specific sights/activities in mind. If you tell us what they are we can suggest others based on your interests.

If you really have no idea - suggest you start with the destinations section on this sight or a couple of good guidebooks - to see what intruiges you most.

Start with looking at the evening activities at Tivoli Gardens.

Hulalady:

I'm at a loss as is Nyt...all I can contribute to your plans without knowing your interests, etc, is a "familiarization" selection of pix. Without a car, you can take a train for short ride to Hillerod and the beautiful Fredericksborg Castle on a lake. Same day, train from there to Humlebaek and the Louisiana Art Museum and Sculpture Gardens. The other two days, there is plenty to see and do within Copenhagen. Look at a guide book and tell us what you've decided to aim at.

Be warned as Nyt said...it's a budget-breaker destination. We visited Norway and Sweden on the same trip. (I figure in about two years, I can get my wife's car back from the bank).
stu

https://picasaweb.google.com/stuarttower/Denmark#

Mahalo (Thank You)!

The Suggestions, Links and Pictures shared above are very helpful & appreciated. I'll see what specific questions I have once I've done more research.

HAPPY NEW YEAR & TRAVELS TO EACH OF YOU!
Hulalady

Enjoy your trip... a few items:

1) We liked the Clarion Collection - Hotel Mayfair. It was 'reasonable' for Copenhagen and included both breakfast and a light dinner (good soup and some small plates... it's buffet, so all you can eat). It's only a couple of blocks from Tivoli Gardens and convenient to the train station to get back to the airport.

2) We really liked the Round Tower. Interesting building and a nice view from the top. Not someplace you'll spend hours, but worth the stop.

3) I'll second the recommendations above. Also, just wander... it's a very walkable city. Have a lunch on Nyhavn. Also the museum pass is a pretty good deal. If you are flying SAS, they give you (or at least used to) a free museum pass (24 hour pass) that you picked up on arrival at the airport.

4) We also liked Rosenborg Castle and gardens.

5) Tivoli at night is great. The exact opposite of amusement parks in the US. Just fun to wander around and have dessert after dinner and check out a little entertainment.

Enjoy the trip!