I have an Australian boyfriend. He is paralysed and wants me to go to Australia. What type of visa should I apply for? And where and how should I start? How long would it take for that type of visa application?

Are you are New Zealander? If so you don't need a visa - you are automatically granted one upon arrival into Australia.

If you live elsewhere, you will need to apply for a Partner visa like this one: http://www.immi.gov.au/migrants/family/partner-outside.htm

You should find all the information you need there to start applying. How long it takes to be processed depends where you are applying. Some offices have less of a backlog than others!