I've never studied Danish. I've studied a few months of Swedish (which is rather similar) and speak OK German, but Danish is not a language I've yet had the time to look at. However, as the wonderful Danish political drama 'Borgen' continues I keep picking up more and more words, which is kinda cool! There's often a debate, especially over at How to learn any language, about if watching TV is helpful for language learning, or is too passive and should only be used as an extra. Indeed, I often agree with this point, although I'm beginning to think that for me I now need more TV than reading, to improve my understanding and vocabulary in all my languages. Here's a short list of words to look out for:

Hej! - Needs no translation
Tak! - Thanks!
Tusind Tak! - Thanks a lot
Undskyld - Sorry
Statsbesøg - State visit
Er du sikker på det? - Are you sure about it?
Jeg ved ikke - I don't know
Jeg vil ikke - I don't want (to)