Bit more detail on where to rent the bike at Central station
I found the bike rental place a bit difficult to locate (from memory) - you walk to the rear end of the sation's central hall (the side with all the shops), not the Tivoli entrance - you'll see some steps downs...walk down the steps, and out into the street (you'll know you're in the right place, as you'll see buildings opposite, not the Tivoli wall and fairground!)
Turn left, and walk maybe 30 metres...there's a side entrance, and inside, there's a long sloping walkway...at the bottom of that you'll find the bike rental.
If you're planning on staying for a few days, and you want to have guaranteed cycling, I recommend a hired bike over the 20DKK free ones...they're hard to find, and are often a bit worse for wear.