I live in australia so i won't be getting my Wii until dec 7th and was wondering, for people that have it, how do you get online with it. I want to do the wireless thing like the ds but what do i use for it? Is it the same adapter the ds uses?

The same adapter should work just fine if you dont have a real wireless access point/router for yoru home.

So if i buy the ds wireless adapter i should be able to use it to connect online with the wii???

I am 90% sure that is correct.

Is it possible to hook into your computer/modem with a usb/network cable?

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The USB has not yet been released and may not be until January:

Nintendo: Online Store - Product Detail