Get a "This Week in Copenhagen" booklet; that will tell you about live music, bars, concerts, museums, and so forth. Chances are there will be a bar with live music within 15 minutes walking distance from your place downtown. If you don't like the bar scene (which is civilized in Copenhagen), ask around at work, or make friends there as irman suggests. But you should know that Danes don't go on dates with strangers the way you may be used to--it is much more common for a group of people from work or from school to go out together, especially if they don't know each other well. Don't ask a co-worker on a date unless you've already gotten to know her well--they won't know what you expect of them. Suggest to stop at a coffee place after work instead, and keep it light. Most likely the woman will prefer to pay for her own coffee/drink. Restaurants are pricey, so "dinner and a movie" is not what people do. Going Dutch is very common--unless things have changed drastically within the last few decades.
Yes, Danes like to speak English, but it is generally a basic form of British English, so be patient and speak slowly! All Danes learn English in school, and the TV channels are full of undubbed American shows, so they are used to hearing American English, but that doesn't mean everyone speaks it (even if they like to think they do!
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You're going to have a great time!