Question: I am traveling from the US to Australia. What type of under the seat bag is practical, and what should be stored in it for the flight?

Answer: 
Under the seat? Hopefully nothing. That's where I store my feet. There's not enough space for a bag.
To add to all the great answers given;

* It can be very cold in the airplane, I like to take a pair of warm socks with me.
* The air can be very dry and rough on the skin, I prefer to take a moisturising cream and lipbalm with me, so I don't start my holiday with itchy, cracked skin and lips.