For a start, even the terms Aboriginal and culture are problematic. Most people involved in this sort of political issue seem to use the word Aboriginal, including the couple of people I met who were of Aboriginal heritage themselves. But a couple of times I got glared at for using it: I think you'll find the word is indigenous. Please be assured I don't intend any disrespect by the term; I'm very much an ignorant outsider here. And as for culture, there are problems with that too. I take erudito's point, but I want to use the most intuitive term I can think of to indicate the kind of things I want to talk about, which I think normally come under the heading of culture.
I noticed that it seems to be trendy to mention Aboriginal issues in every possible context. I'm not saying that's a bad thing; it's a good counter to invisibility issues and so on. But I didn't really know what to make of things like random plaques with Aboriginal stories on them. For example, the synagogue I went to has a banner on all its handouts saying "This community would like to acknowledge that members of the Kulin nation are the traditional custodians of the land our syngogue stands on". Which is all very well, but it's hard to see how it actually does anybody any good!
I did set out with the intention of educating myself about Aboriginal culture, though, and with all these resources available I did indeed learn something. The Immigration museum is very good indeed; I would say one of the best museums I've ever been to. Obviously as the name would suggest it's not directly a museum about Aboriginal issues, but it did give me a context for the history of relations with Aboriginal peoples. I sort of knew about the early settlers regarding Australia as Terra nullis and treating the indigenous population very badly, though the museum filled in a lot of the horrifying details for me. What I hadn't been aware of was the White Australia policy which continued well into the 20th century. I've always been mystified as to how there could be racism in a country like Australia which is so much built on immigration, and the Immigration museum explained this all too clearly. (There are of course repercussions for non-European immigrants and their descendants, as well as injustices towards Aborigines.)
I also made a point of going to the Indigenous Collection at the main art gallery, the National Gallery of Victoria, on the basis that I wanted to see art different from the sort of thing I can see in any old European gallery. (That's where I got in trouble for using the word Aboriginal; to me, the term indigenous art doesn't make sense, because the art isn't native to the country, it's something that an artist has created.) There's always the difficulty of presenting this kind of thing so that it's art rather than anthropology of "primitive" cultural artefacts; I'm coming to the conclusion that the most effective way to do this is to include the name of the artist wherever possible. Refusing to give any explanation of the significance of the works within the culture they come from doesn't help.
The NGV has absolutely terrible labelling full of pretentious wank that doesn't convey any useful information about the artworks, but at least the Indigenous collection got the same terrible labelling as the rest of the art! Personally, I found the Aboriginal art quite difficult; it's very abstract, and full of conventions that I'm not aware of, so I don't really have the tools to look at it properly. But I got something out of the collection; some of the paintings really convey movement and to some extent emotion, even to me as someone not at all versed in the artistic conventions.
For me, the most informative part of the trip was an Aboriginal Heritage tour of the Botanic Gardens. The guide was the sort of person I would find very annoying if he came from my own culture; he was very into "spiritual" stuff and generally a bit tree-huggy. But his material was very interesting, and very... personal, direct, he wasn't embarrassed to talk about his own life and experiences as an Aboriginal in modern Australia. Most of the tour was fairly standard stuff about how the Aborigines were really in touch with nature and had a lot of knowledge about how to make use of all the plants and so on in the environment. But the personal glimpses made it really worthwhile.
The guide also pointed out what erudito said in his comment, that there wasn't one "Aboriginal culture", but several hundred, not only with distinct customs but distinct languages. And all that is now reduced to cute little staged "ceremonies" and "Dreamtime" paintings and placards for tourists, and kangaroos and boomerangs (the other Aboriginal guy I met was at the wildlife sanctuary, doing a sort of little stage show where the humour consisted of his being really bad at throwing boomerangs, which was cringey to start with and went on far too long repeating the same joke). It made me realize how profoundly I disagree with rysmiel's view that humans are a renewable resource. I mean, a cynic could argue that Aboriginal cultures didn't contribute much to civilization; none of them got as far as agriculture, let alone writing. But they had ritual burial a minimum of 60,000 years ago, and several hundred ancient human cultures were destroyed, mostly deliberately, and that's simply not replaceable, even if you don't bring morality into the question at all.
Greater minds than mine have tried to deal with the question of whether there's anything meaningful that can be done now, since this has happened and history can't be changed. I suppose having all the "Aboriginal myth" plaques all over the place is better than nothing. It's absolutely a given that racism against people of Aboriginal descent is completely inexcusable, but simple anti-racism doesn't seem to go far enough here. There isn't really a way to give everybody the rights of modern, essentially western civilization while at the same time preserving the Aboriginal traditions. At least not preserving them as living traditions rather than museum pieces and tourist attractions. It's probably too late, actually. There have been too many people killed, not to mention deliberate and largely succesful campaigns to destroy cultural elements such as language teaching and religion. (And it's not like this kind of thing is unique in human history; I am aware that there are far too many comparable examples.)