Every Danish person we spoke to whether it was in hotels, shops, restaurants or on the street, spoke perfect English. After our first day in the city we stopped asking people if they speak English when asking a question or beginning a conversation - rather presumptous on our part but there was no point in asking just for the sake of it.
A good tip is to learn a few phrases in the local language of the country that you are visiting. For me Danish is a hard language (I would say it seems a bit harder to pronounce than say Swedish or Norwegian, even though the three languages are somewhat similar).
But here are some basic phrases taken from my Lonely Planet Scandinavian Phrasebook that you may find helpful:
Hello/Goodbye (informal): Hej / Hej hej
Hello/ Goodbye (formal): Goddag/Farvel
Thank You: Tak
Do you speak English?: Taler De engelsk?
I (don't) understand: Jeg forstar (ikke)
Free Admission: Gratis Agang
Danes are proud of a few things: their flag, some of their most well-known sights, their beer, their royal family....and a word: hyggeligt. They say that this word does not exist in any other language, and that it cannot be explained. In fact, it does exist in other languages, as the very same word is used in Norwegian, Swedes have a different word with the same meaning, and if I'm not mistaken, the Dutch have a similar word with the same meaning too. But, ok, I suppose the big world languages don't. The closest synonym in English is probably cozy, but 'hyggeligt' means more than that. A person can be it (warm, friendly, making you feel at ease), a house can be it (somewhere you'd curl up in front of a fireplace with a warm fuzzy feeling inside), a dinner party can be it......anything can be hyggelig(t), really. It can be both an adjective, an adverb and a verb. So now you know...
Another language curiosity: In the english language danish pastry is called...well just that: danish pastry. In denmark, though, it's called wienerbr?d, as in "bread from vienna". Anyone know what it's called in Vienna?
It was amazing to us that our Swedish and Danish friends had to speak English to each other, as the two Scandinavian languages sounded so similar to our untrained ears. They also had quite a rivalry and argued ,though good-naturedly, the whole day!