I know this is an old post, but I wouldnt say there is any specific degree needed for politics. Law and Political Science are probably the most relevant, but I dont think they are necessary. Im not too sure regarding the specifics of entering politics (i.e. dealing with the AEC etc), but politics is all about who you know. If you dont know someone influential who can shortcut you in, then the best way is to join a major party and get involved with the local branch (as mentioned above). It is pretty boring at first, but it's all about meeting people and making an impression. That could lead to many opportunities, maybe as a staffer to a politician or being an organiser. As I said, it's all about building connections and moving upward.
Going into it as an independent is much more onerous. If you are interested in the lower house in particular, you will struggle as most people tend to vote major parties. Independents have more luck in niche electorates (the bush, inner city etc). You also need a lot more name recognition, which is difficult to obtain.