I don't fit the criteria you mentioned, but I'm fairly close as a Canadian with the option to obtain UK Citizenship whenever I feel like, I just have to do the forms. I'm not a lawyer or anything, so bear in mind that this is all just based on what I could find with Google, so take it with as much or as little salt as you want.
There doesn't seem to be anything stopping you getting the Australian citizenship with regards to the laws of either country. Both countries appear to completely tolerate multiple nationality. There is a decent Wikipedia article on US nationality law with regards to dual citizenship. Basically unless you're planning on joining the armed forces or applying for a US security clearance, it sounds as though you'll be fine.
Also, I don't think you're actually under any obligation whatsoever to inform the US government that you have become a naturalised citizen of another country, unless that country requires you to formally renounce your US citizenship in order to obtain theirs.
Now regarding travel between the two countries, you should always bring both passports and enter each respective country on their passport. Australian citizens are required by law to enter and exit Australia using their Australian passports. I'm not aware of a similar requirement for the US, but I'm sure it'd be in your best interest to use your US passport. If you ever get quizzed about it in either country because of confusing passport stamps (like an exit without an entry or something) just be honest and say you're a dual national. It's certainly not illegal, not to mention fairly common these days.
A friend of mine with dual Canadian/Polish nationality traveled to Poland once on her Canadian passport and didn't get into any trouble, despite her place of birth on the passport as a town in Poland. I was a little surprised by this, but perhaps the border guards didn't notice or care.
You will be able to answer this for yourself, as I'm not sure what it's like for non-resident US citizens returning for a holiday or visit, but in Canada we have to fill out a landing card and put down whether we're visiting or a returning resident. If I put down I'm a returning resident, I'd very likely cause myself a big tax and residency headache if anyone caught on. The same might apply for you in either the US or Australia depending whether they also have landing cards, what kinds of questions they ask, etc.
Hope that helps!