You say your partner ... you will need to be married to make a joint application for immigration to Australia. Normally, a cousin cannot sponsor your application unless you (and your spouse) have no brothers, sisters, non-dependent children, parents (or step-equivalents) other than those in Australia. Unless this is the case, you will need to qualify for immigration in your own right.
Unless you are the remaining parent of an Australian citizen, there is no way to fast track this process. See this link for details about applying:http://www.immi.gov.au/immigration/
I see from your comments below that you claim to be an Australian citizen yourself. If so, you might be able to apply for an immigration visa as returning resident. to qualify though, you need to have spent at least nine of your first 18 years in Australia as a permanent resident. Again, for your partner to be included in your application, they must be your spouse. You can follow the Migrants > Permanent Residents links from the above Website.