Hi Sara!
Yes, Tivoli will be one recommended stop. You should also make sure to check out Nyhavn, a wonderful waterfront place to relax and grab a drink and a bite. Also, take a stroll along Langelinie, the pier where the biggest cruise ships dock. And if you want to see something very special to Denmark, show up at Sunday noon on the center court of Amalienborg, the Queen’s palace—you’ll see the changing of the guards!

There’s so much more to see, I hardly know where to begin. Take note of the small details, too: have you ever been in a city with so many bicycles?

Cheers from Austria (but I’m really from Copenhagen!)
- TorbenGB

Hi Sarah, I lived in this beautiful city for 4 years… to take a slightly quirky angle to that of Torben(very good tips mate!) When you get to Nyhavn take the boat tour round the cannals and mind your head on the low bridges – best way to relax and see the city from a whole different side. Food : You got to eat a ristet hotdog from a hotdog stand with all the trimmings. Drink: You have to try a scnapps(preferably with karry sild(herring))and wash it down with Carslberg. Pub: 1)Grungey; The Moose on Sværtegade 5 – open all night(6am) anything goes bar. 2)Live Music; The Drop Inn, Kompagnistræde 34 get in before 11- excellent free gigs. 3)Posh http://www.sommersko.dk/ – I rarely went there. Outside the city its real nice to take bus to Dragor(kinda conservation village and seaside harbour 30 mins drive). Just walking around the city and stopping off for coffees and so on is the one thing I miss about Copenhagen. The cafes are simply too damm cool. Enjoy. Chris

I think Nyhavn is turisty and pricey, but still it’s worth seeing. Buy a few beers in a store and sit down by the water and look at all the people. Warning, a lot of other people are also doing this, so late nights it’s crowded with drunk youth.
I love Tivoli, it has an old feeling to it. You should go after dark :)
My personal favorites are Louisiana, museum of modern art (it is a truly beautyful location by the water, you’ll have to take a train there ) and Glypoteket (right next to Tivoli, a silent oasis with lots of statues and mummies!) Check out: http://www.louisiana.dk/uk and http://www.glyptoteket.dk/
You should also take a stroll by the lakes, I don’t know how it is in Christiania now, but it used to be a different experience! http://www.christiania.org/modules.php?name=NukeWrap&page=/inc/guide/

My shopping favorites :)
“Illum Bolighus” (heaven for lovers of scandinavian design), “Istedgade” (the first blocks close to the train station aren’t nice, but keep on walking! Cosy cafes and nice boutiqes will appear) and I like “Købmagergade”, a sidestreet to the crowded “Strøget”

Have a nice trip!

I was there for the first time a few weeks ago. I highly recommend having lunch at the Soren K Restaurant (named after Kierkegaard) at the Royal Library. http://www.frommers.com/destinations/copenhagen/D45699.html

The Hotel Alexandra was very nice, with free Internet (although you have to pick up free vouchers at the front desk for 3 hours apiece) and great location near the Tivoli, but be sure to get good directions to the hotel— don’t rely upon Google Maps!

I also highly recommend taking the short train ride to Malmo, Sweden, which is a lovely & charming little city. The Elite Hotel Savoy was a wonderful place to stay, and the location was perfect— right across from the central train station, and a block from the main squares.

Have a great trip!