I stayed at Hotel Alexandra (H.C Andersens Boulevard) which was a good central part of town for the first time Copenhagener. It's also excellent as it is filled with mid century Danish furniture.
No matter where you are staying, hire a bike. But hire one that is in your size. I didn't and it meant I felt like I was jumping on a horse every time I got on it. Also: road rules. There are many many rules for bikers. This was a bit tricky for a gal who is used to zero road rules China style. Study up.
Copenhagen has an overwhelming amount of food choice. Torvehallerne food hall is a great place to eat lunch, pretty much every day. Mirabelle bakery is (Guldergsgade 29) has amazing bread and wonderful breakfasts. I'd stay near there next time and eat their breakfast plate every day and try their other restaurant Baest next door for home cured meats and other deliciousness. Gasoline Grill was a standout find - burgers served in an old service station. At night I ate at Kodbyens Fiskebar and the food choice around Kodbyen look amazing. Uformel (B Formel edgier younger brother) was also a standout choice. More about food later.
For cocktails I went to Ruby and Lidkoeb. Lidkoeb was a lovely dark setting with interesting cocktail options and great service. Ruby had a lovely outlook over a quiet street and canal where I watched the late night traffic and dog walkers from my seat at the bay window.
Things I did: Design Museum Denmark, Rundetarn for great city views, Louisiana Museum of Modern Art, the little mermaid (because you have to) and generally cycling around (because all the building and lakes and rivers are beautiful).
Shops - look up: Limited Works and Paper Collective (for original art), Ravenborggade street for antiques, Cinnober for beautiful candles, books and stationery. Tiger for well designed bargains.