Two sites, still in Beta, provide a quick and excellent overview of bike lanes. Cyclecopenhagen.dk details the quickest, greenest, safest or quietest route directions (which you can print) to your destination in Copenhagen. Cyclistic.dk provides touristic points of interest on a larger country scale.
Or go old school and download Copenhagen City’s Cycling Map Copenhagen PDF and save it on your smartphone for offline use. A print version can be picked up at the tourist information.