The Opera House website has a section with detailed instructions and small maps.
Harbour Bus 
The harbour buses have a shuttle service in connection with performances on the Main Stage of the Opera House. Departures are from the Nyhavn Canal outlet into the harbour, right by Hotel 71 Nyhavn.
Boat lines are 901 and 902. Roughly they sail every twenty minutes, but it is a good idea to check at the pier.
Timetable (Danish only)
Water taxi 
Copenhagen Water Taxi sails to and from the Opera Houses. There is a shuttle service from the pier in front of the Hotel Admiral one hour before and one hour after a performance. Or if you are 8-10 persons, you can order your own water taxi. Read more on www.flyvefisk.dk
By bus
Bus No. 66 goes straight to the Opera House. After performances on the Main Stage there are extra busses to the Central Station. via Christianshavns Torv (i.e. Christianshavn Square) and Christiansborg Palace.
Timetable (Danish only)
Metro
The Metro stops at Christianshavns Torv, where you can catch bus No 66 or walk about one kilometer to the Opera House.
On bike or foot
Turn down Prinsessegade from Christianshavns Torv and head towards Holmen.
Parking
There are several places, where you can park on Holmen. A limited number of cars can park on the parking lot just north of the Opera House - it is free from 3pm to 2am on weekdays and over weekends. But during the day a parking fee is charged. Please check the signs, to avoid any disagreable suprises!
Parking space for persons
Four parking spaces have been reserved for persons and three larger spaces for special buses for .
Read more here