I don't know what it's like in the USA, but in Australia it's true to an extent.
Generally speaking, immigration to Australia has come in waves.
First you have the various Aboriginal nations, here for 60,000 years. Those guys do have it very rough in Australia, no question at all. There is an inherent thing you hear in all sorts of towns when travelling to "watch out for them" as if they are all up to no good.
Other migrant groups we have include (very early on in Australia's colonisation) Europeans aside from just British, Chinese, Afgans who were all here in the gold/exploration era.
Post WW1/2 there have been a lot of German, Italian, Greek, Turkish immigrants who have settled in seamlessly. Except the Italians and Greeks who seem to love having a go at each other every major football tournament but that is a good thing. And also a lot of people from Pacific Islands too.
In the last thirty odd years there has been a lot of migration from South East Asia, notably Malaysia, Singapore, Philippines, Vietnam etc. And a continuing influx of people from China and Japan, as well as India, Pakistan etc.
There has also been a big increase in immigrants from the Middle East and Africa, notably Lebanon and Sudan.
Probably the groups that have the worst of it in Australia would have to be the Sudanese and Middle East immigrants, but I tend to believe that it is only a matter of time before on the whole that is a non-issue. As far as I have experienced, most people tend to see 2nd generation or older people as "Aussies". Chinese/Italian/Greek etc migrant families are so ingrained in this country now that pretty much forms the model for how all the new groups will hopefully fit in.
It takes longer to settle in and be accepted than it should, but I reckon in 50 years, (hopefully much sooner), African and M-E immigrants won't even be noticed any more. They'll just be as accepted as Italians and Greeks are, for example - ie, completely.
But the worst problem I feel will remain the treatment of Aboriginal people. There is an attitude for change amongst Australians, but its very slow and there is a pervading sense of "how do we look after them" about it, and I imagine that must be very patronising for them. Especially with such things as the intervention in the NT that are seen in very polarising ways all over the country.
If you are white in Australia though, there is pretty much no racism directed your way except in the presence of small, tight knit immigrant groups that have a very strong "us and them" identity. But none of that will harm a white Australian's chances of getting work etc.
I don't believe that there is widespread racism here in general, but there are some groups that do have it noticeably worse than others. I guess the key is to recognise it is there, and try to deal with it. I think the true measure of Australia's maturity in dealing with it will be to bring about a practical reconciliation with Aboriginal nations rather than just stopping at the symbolic one a few years ago.