Without a doubt the safest way is to purchase your mobile phone in Australia. If anything goes wrong with the handset, you will have a local contact point to get it repaired under warranty. You don't want to end up spending a fortune shipping it overseas and back.
That said, GSM networks in Australia operate on the 900/1800MHz spectrum, which is compatible with most networks in the UK, Europe and Asia.
So ensure any phone that you purchase is, at minimum, a dual-band handset supporting the GSM 900/1800 standard. As a rule of thumb, steer clear of phones from the US and Canada. North American networks are generally based on the GSM 850/1900 standard. However, if you plan on frequently travelling overseas, look for a tri-band (GSM 900/1800/1900) or quad-band (GSM 850/900/1800/1900) handset for maximum flexibility. For more information on the different networks in overseas countries, take a look at the GSM Association's Web site.
A word of warning before you make your purchase, ensure the phone is not SIM locked. Mobile phone providers often lock handsets that are given out free of charge on a contract to only accept SIM cards from their own network.
Finally, if you are buying your phone though an online auction site such as eBay, make sure to check the feedback in the seller's profile before placing a bid.