Danes speak excellent English, so there's no need to learn Danish if you're staying in the Copenhagen area. But it doesn't hurt to try!
Danish is the official language of Denmark (not Dutch!!), but all Danes learn English in school and most Danes, especially those in the Copenhagen region, are quite comfortable speaking English. The first time I came to Copenhagen, I remember making the mistake of asking a few Danes, "Do you speak English?" or, "Is it okay if I speak English?" and was always met with a strange look and, "Yes, of course....?" So... I stopped asking.
It's worth noting that if you are planning to venture outside of Copenhagen, people might not be so comfortable speaking English simply because they might not be around tourists as often. But they will still understand and generally be quite gracious about it, assuming you are kind and polite in your interactions :)
That said, it certainly doesn't hurt to learn a few Danish phrases before you visit!
Hello: hej (pronounced "hi")
Goodbye: hej hej (pronounced "hi hi")
Yes: ja (pronounced "yah")
No: nej (rhymes with "hej")
Please: There actually is no word for please! But if you are offered something, like water at a restaurant, you could say "ja, tak" (yes, thank you) instead of "yes please."
Thank you: tak (pronounced like if you were saying "tock," but with your mouth shaped like you're saying "tack." So the vowel is somewhere between an English short "o" and short "a")
You should also be aware that Danish is not pronounced like it looks to English speakers. If you need help finding something, you might have better luck showing a Dane your phone with the name of the place on it, rather than trying to guess how to pronounce it ;)
Something like Duolingo won't get you far if you want to seriously learn Danish, but I think it can be nice for getting a little intro to the language and how it sounds!
Vai a consultare la mia guida per le migliori cose gratuite da fare, nonché itinerari e consigli di viaggio per rendere il tuo viaggio indimenticabile.