If someone is flying from the US to a city in Australia and has to take a connecting flight in a different state, e.g. flying from Dallas to Darwin in the Northern Territory, and they have to take a connection in Sydney, New South Wales, then is the custom check only at the Port Of Entry, e.g. Sydney? Or since different states in Australia have different Customs rules, will there be a need to clear Customs in Darwin too?
Different states do not have different Customs rules. There are some special zones around Australia but those aren't enforced by Customs. When catching a connecting flight it depends on whether your connection is departing from the international or domestic terminals.
Unlike many or all US airports, Australian airports that regularly service international flights have their international operations completely separated from their domestic operations. This means that if the connection you're intending to catch, e.g. Sydney to Darwin, is leaving from the international terminal you will only have to clear Customs upon arrival in Darwin. However if it's leaving from the domestic terminal, as is most likely the case, you have to clear Customs in Sydney. You will only clear Customs once.
When dealing with the special zones regarding transporting of fruit, etc, within Australia, there will be prompts and bins to discard the foodstuffs on your way out of the airport but there will not be Customs-style checks to verify you don't actually have any.