There are a lot of free things to do in Copenhagen, and there are a lot of things you can do at very little cost and many cheap alternatives. Cruise the harbour on a harbour bus instead of taking a canal tour. Take a city bike instead of renting a bike. Eat for DKK 50 at Copenhagen Street Food. And if you do not mind standing up during a performance, you can also find very cheap tickets for The Royal Danish Opera from DKK 75-125.
Visit Bakken amusement park instead of Tivoli Gardens. Bakken has no entrance fee, you can bring your own food and drinks, and if you want to go on the rides, a multi-ride ticket costs DKK 249, or DKK 179 for kids' rides. If you visit on a Wednesday, you get up to 50 % discount when you pay by cash on the rides. So if you are visiting Copenhagen on a budget, there is no need for you to not experience the city in full.
Remember that a Copenhagen Card gives you free admission to more than 70 museums and attractions, free public transport by bus, train and Metro and discounts on restaurants, rentals and sights. You can also bring along two children under 10 for free.