Best neighbourhoods & craigslist apts (Copenhagen)

I'm relocating from New York to Brande (near Herning) for work and would like to rent a place in Copenhagen for the weekends. Any suggestions on the best neighbourhood to live in? I would love to be near good running (park or waterfront), cool cafes, bookstores, art galleries...

I started looking at places on craigslist, but came across an ad saying that most of the postings were fake - has anyone else had experience with craigslist in Copenhagen?

Theresa