Can I marry my same-sex partner?

No, this is not permitted.  The Marriage Act 1961 is very clear in stipulating that a marriage is "... the union of a man and a woman to the exclusion of all others ...".

However, it is possible for a Marriage Celebrant to act as a generalist celebrant and conduct a love commitment ceremony for you and your partner.  This is permitted provided the ceremony does not purport to be a legal marriage.   In such a case, your Marriage Celebrant would prepare a Certificate of Commitment for you and your partner to be signed by yourselves and your witnesses during the ceremony.

What is the situation in cases where there has been a change of gender?

There is now a case law precedent (arising from the case of "Kevin" heard in the Family Court of Australia) to the effect that a post operative trans-sexual female to a male person is a "man" for the purposes of the Marriage Act 1961 and as such, is able to enter into a valid marriage with a woman. 

The decision of the Family Court does not affect the definition of marriage in Australia - it remains as "... the union of a man and a woman to the exclusion of all others ...".  The parties to a valid marriage must be a man and a woman.  The issue in the case was the status to be assigned to "Kevin" who had undergone sexual reassignment surgery and had been issued with a new birth certificate in New South Wales to reflect his reassignment as a male.  The decision is only applicable in a post-operative trans-sexual situation.

This decision means there are circumstances in which a person who has undergone sexual reassignment surgery may be able to marry in their reassigned gender as long as they fall within the parameters set down in the decision in the case of “Kevin“.