First of all, don't worry about being taxed twice. Australia and the USA have a tax treaty that avoids double taxation for ordinary taxpayers. You do however need to know that all US citizens are required to file taxes every year regardless of where they live or where they earn their income. As a student and very low (if any) income earner, at this stage of your life, you only need to notify the US IRS that you don't earn enough to file a return. Have a word to a tax accountant who knows about international taxes if you're not sure what to do. It will only ever become important if you want to live in the USA at any time, but at your age, it's wise to keep your options open by doing the right thing now.
You'll need to get a SS number, but it's very easy to get one.
And take no notice of anyone who tries to tell you that you are not a US citizen or that you had to do anything when you turned 18 to keep that status. You're a 'natural born' US citizen for life and could even run for President if you wish to do so (though only from age 35 and only if you've lived in the USA for at least 14 years).
A downside to dual citizenship is the tax filing requirement, but your mother is right in advising you that you should never renounce your US citizenship. It's hard to get for most people and could well be extremely valuable in the future if you ever want to work there in order to get marketable international experience.
I'm also a dual US and Australian citizen