International relations-wise, the most obvious career is working in the diplomatic service. However, this is very competitive - DFAT's primary intake of graduates has about 25 places a year, and something like 2000 people apply. (There are other intakes for Aborigines/Torres Strait Islanders and for Corporate/Management graduates.) Think-tanks - privately funded research institutes - also have graduate programs and internships. There aren't many international policy think-tanks in Australia, as far as I know, but there are a ton in the USA and to a lesser extent Britain if you're willing to move overseas.
Other internationally-focused employers are multilateral agencies like the UN, the International Criminal Court, the International Court of Justice, the World Health Organisation, the International Seabed Authority and all the rest. There are millions. Bear in mind that you need different qualifications for some of them: the ICJ, ICC and ISA would probably be more interested in graduates of international law, rather than international relations, the WHO would probably like people who know something about public health, the qualifications you need for the UN vary depending on what position you're applying for. But I suspect an international relations major would be a good start.
NGOs (non-governmental organisations, in the international context usually charitable or environmental or whatever) also have internationally-focused jobs. Again, qualifications vary, but international relations graduates would probably be looking at policy formulation or research jobs.
Apart from all those, companies operating in the international environment might be looking for risk advisors, political analysts and so on. Don't know if they'd want people with commercial qualifications as well for those.
Anyone want to add to that or give some ideas for political science grads?