If you applied for your Australia Tourist visa online, then you would have first created an Immi Account to do so.
Immi Account can be found here: https://online.immi.gov.au
Log back into your Immi Account to see how your Australian tourist visa application is tracking.
You should receive emails directly from the Department upon lodgement and decision. If the Department requires more information from you before they can make a decision, then they will also directly email you (or your Registered Migration Agent) a Request for Further Information.
If your visa has already been granted, you can check your visa details and visa conditions via VEVO: https://immi.homeaffairs.gov.au/visas/already-have-a-visa/check-visa-details-and-conditions/check-conditions-online/visa-holders
Why is my Australian tourist visa taking so long?
There can be many reasons that your visa is taking awhile. It may have been caught up in the system, it might be a particularly busy time for the Department or there may be more information that the case officer requires before they can make a decision.
The Department does not have to process your visa application in any particular time frame and they definitely don’t respond to requests for updates on processing times. The indicate times frames published should give you an idea of what you can expect, but otherwise it is best to keep an eye on your Immi Account.