We am flying from Australia to Rome. A couple of days later we fly across to Athens then back to Rome several days later. Do we go through customs each of those times?
Yes. In each case you have to walk through customs to enter into the arrivals terminal on the 'landside' of the airport. As you are travelling from one EU country to another you simply walk through the green 'nothing to declare' section.
To elaborate on Al's comment, when you land in Rome, there will be an immigration/Passport check (a line), customs is like Al said, if you need to declare, otherwise you walk right out, no line. Upon landing in Athens, you walk right out of the airport, no line; same with back into Rome. I did not notice much in the way of Customs between Schengen/EU countries unless you were arriving from an international (Non-Schengen/EU) flight, though there may have been an office to declare any goods if needed.
Arriving from ouside EU, you will go through customs after retrieving luggage. Use the channel saying "arrival from outside EU, nothing to declare" (usually the green channel).
Between EU countries, Theoretically, yes, but between EU countries, customs booth are often unmanned or there is one person wawing everybody through. Use the channel saying "arriving from other EU country", often the blue channel. Furthermore, travelling between EU countries, there are virtually no restricitions on what you can bring (except illegal substances, narcotics etc.)
There are no immigration checks when travelling between Schengen countries.