Hi Joseph,

I'm sure that there are many online brokerages in Australia that will let you buy mutual funds or ETFs (Exchange Traded Funds)
that are U.S. based, among the popular ones:

Vanguard Total Stock Market Index Fund ETF Shares (VTI)
Vanguard 500 Index Fund ETF Class (VOO)
PowerShares QQQ

I don't recommend that you buy individual stocks until you've accumulated enough assets to be able to diversify your holding.
No, you don't need a US bank account. But you'll have to pay a premium for investing internationally in the form of higher
expense ratios. I suggest that you don't limit your investing to just US-based stocks. US market performance may be leading,
but may not be true in the future. The Australian market is just as healthy.

As far as taxes is concerned, please refer to the following link:

https://www.ato.gov.au/Individuals/International-tax-for-individuals/Investing-in-Australia/

As with most developed countries, you will have to pay capital gains tax when you sell the stock with a profit and income tax
from dividends that you receive on a periodic basis.

Good luck. I hope you achieve your goal!