Anywhere between the Victoria/NSW border and the NSW/Queensland border, just head east and you will hit a beach and you will be able to surf it (well, 90% of them). Admittedly on many you will be by yourself (and wont be able to hire a board nearby).
How good a surfer are you? Sydney northern beaches (Manly up to Narrbeen) are all good, but the further north you get the more tribal it becomes. Manly is fine for anyone from tourists discovering waves for the first time to good surfers although it can be busy. If you head further north, particularly if you arent good ('Australian good'), then you may find it more enjoyable to surf on the 2nd/smaller peaks than to try and mix it with the locals (its not terrible, but you dont want to deal with it if you are happy with taking the smaller stuff - you wont get beaten up or even abused, you will just get dropped in and snaked and so forth if you arent up to scratch)
In Sydney drop by the surf shops in Bondi and Manly and you will be able to rent a board at most of them. They will also give you some info about where to go.
Aust beaches are just as prone to any other beaches to have periods of no swell so dont think its all going to be magic. Dont forget the sunscreen! Around end of Jan you probably can get by without a wetsuit for an hour or so (water temp will be around 20/21C); might be worthwhile taking along a 1mm or 2mm top if you have one to give yourself a bit of leeway or to block the wind. Then again, if its 43C like the other day...
I don't know around Melbourne. As mentioned, its not actually on a surf beach so you will have to travel a little bit out of the city. There are trains. The water will be colder. Also - dont look for surf in Geelong (@roberdy!). You need to travel a bit further