Although it’s named Canberra International Airport, very few flights actually serve destinations abroad. But there are numerous flights from Melbourne, Sydney and Brisbane.
Apart from the bus from Sydney (murrays.com.au) which takes just three hours, it’s a long journey by bus or train from Sydney or Melbourne. The train from Sydney takes 4 ½ hours (go figure) while the journey from Melbourne takes forever.
Canberra’s bus network, ironically named Action, serves the whole city. Services run on time (mostly) but the big problem is that services are infrequent. You could wait 30-40 minutes for a bus on some routes. A single trip (valid for 90 minutes) is $3, but a book of 10 tickets will cost $22. Tickets can be bought from the driver, or from newsagents.
Canberra is a big place, geographically speaking. The attractions are spread out, making walking a hassle. Combine this with infrequent buses, and a car is going to be your best bet, by far, for getting around. Being a planned and well-funded city, the roads are very good. But they’re not laid out in a grid fashion, meaning you’ll definitely need a map or satnav.