Not sure this is a beginner thread.
If trading means live buy/sell on the ASX(day trader style) , no, you can not do this after hours.
But if you mean, can I place buy/sell order in the evening after analysing the day movement, yes you can: you just need an online trading account (comsec, bell direct, etc etc) and then place your orders: they will be executed the following day at the open, or even in the case of conditional orders, after a specific price, volume or time trigger.
But you do not indeed need to call a person during office hours to trade the asx.
Few people have that luxury
Hope it helps