The bank I am with is ME Bank, which has the best interest rate on the market that I can find, and doesn’t invest in any fossil fuels. I used Market Forces to find out about them and compare with other banks. However, interest rates are pretty low, so I have moved most of my savings to a mutual fund, called Australian Ethical Investment. My funds are in an emerging companies funds, which offer a high rate of return with a higher risk (but the risk-adjusted-return metrics seem to be more favourable).
Find better deals for other things you pay bills for such as your mobile phone provider and your internet service provider. My mobile phone provider is Yomojo, which will recommend the best plan for you based on your past usage. You can adjust the amount of voice, text and data from a monthly usage amount with a fee or pay as you go. Shop around for a bank that provides high-interest savings accounts and term deposits. Extra features may be useful (depending on your circumstances) such as online banking; no exit-fees, lock-in contracts, minimum deposits and so on.
Money saver programs with matched funds for savings
The Australian Securities and Investments Commission (ASIC) MoneySmart page gives many tips on how to manage and save money. Simple savings tips on that website are here. I think I have covered all the tips on saving energy and water. Other tips on saving on groceries, saving money with a friend, clothing and other items, wedding, banking and habits are provided.
The site also links to a Saver Plus program that gives you $500 for saving $500 over 10 months with $50 minimum monthly deposits. I have signed up for it. To be eligible, you need to have a Centrelink Health Care Card or Pensioner Concession Card, have a regular income, be at least 18 years old, have a child (5-18 years old) or attend vocational education yourself, and the money that they match is used on educational expenses. You deposit a minimum of $50 a month for 10 months into an ANZ savings account which has a 3.7% interest rate. After the end of the 10 months you get the extra $500, which you must use it for educational purposes.