What time of year are you thinking of going? Do you have particular interests? Museums, castles, etc. We absolutely loved Copenhagen - I've no doubt you will too.
Depending on your family's interests, I think the National Museum is well worth a few hours. Very interesting historic collection. Rosenborg castle (and gardens/Kongens Have) are also terrific, and on a nice day, a great place for a picnic. Tivoli (an old, historic amusement park) is one of the top tourist spots in Copenhagen. We didn't go there (it was closed during our trip), but many love it. If you like art, both Thorvalden's Museum (sculpture) and the Ny Carlsberg Glyptotek (mostly paintings) are excellent. Freetown Christiania is not for everyone, but we found it fascinating. Read up on it and see if it interests you - I won't try to sway you with my description. But if you go, definitely walk beyond the main street/area and see some of the more unique houses and the quieter parts, which are beautiful. We also went to the Amalienborg palace, which is more about the history of the Danish monarchy than anything else. I liked Rosenborg more, but Amalienborg is nice too. Many people go to the Little Mermaid statute - we did, and it is tiny and not particularly impressive. If you are out that way (we were), then stop by and snap a photo, but don't go out of your way for it.
I'd also recommend doing a canal tour (I think they are like an hour or 90 minutes long). It's a great way to see a lot of the city.
As far as saving money, we purchased the Copenhagen Card. Its value depends on (a) how many of the included activities/places of interest you do and (b) how much public transportation you will use. We did several day trips out of the city (to Roskilde, Helsingor, and Hillerod), and all of our transportation costs were covered by the card. It includes all transportation both in Copenhagen and almost everywhere on the island of Zealand. It also covers a lot (but not all) of the museums and places of interest in Copenhagen. The website (in English) is very thorough and helps break down your savings. But it isn't guaranteed savings either - you have to check to make sure it's worthwhile for your itinerary.
There are lots of beautiful places to wander and explore. In addition to the Rosenborg gardens and the quieter sections of Christiania, definitely check out Nyhavn, the Orstedparken, and walking along the many canals.
In terms of restaurants, pay attention because Copenhagen is an expensive city in which to dine out. I can't recommend Relae enough. It was recently voted as the #40 restaurant in the world, and carries a Michelin star. When we were there, we got a 4 course meal for approximately $68 (USD). It was one of the best restaurants I've ever been to. It has a "cheaper" (relative) bistro across the street called Manfreds og Vin. We weren't impressed, but it has consistently great reviews, so maybe we caught them on a bad night. The Cofoco chain has some good restaurants that are more affordable (we spent probably $100 for two people with one drink each). Note that a lot of restaurants in Copenhagen charge for water. Last, I would definitely recommend going to Copenhagen Street Food, a food court on Papiroen (an island, in Christianshavn). It has a couple dozen different vendors with every type of food imaginable, and it is situated on a canal, with plenty of outdoor seating.
We aren't big shoppers, so don't have much feedback there. The Stroget (a long, car-free street) is full of high end stores. We walked along it mostly to people watch.
I'm sure I am forgetting plenty. Send me a PM if you have any questions. There's also a local on here (Morten), and he knows way more than I do.