You can exchange money at any one of the numerous banks situated in the city centre as well as around the city. There are plenty of ATM's located around the city, and you will usually find them just outside of the bank.
There is also an American Express in the Centrepoint building, (corner Petrie Street and City Walk), and a Thomas Cook in the Canberra Centre, Bunda Street. If you need cash out of normal opening hours then try the Casino Canberra.
At Canberra Airport there are also some foreign currency exchange outlets and ATM cashpoints.