Yes! It’s possible to travel around Australia by public transport. There are numerous ways to do the trip by using trains and coach services and numerous routes you can take.
Sydney – Darwin – Broome
We’ll start the trip by heading north from Sydney up to Darwin and over to Broome in W.A. using Greyhound coaches. Greyhound offers travel passes starting from $329 for a 15-day pass to $1,699 for a 365-day pass. Full details of these passes are at Whimit Travel Passes
You can also do part of the journey by train, from Sydney to Mt. Isa, then Greyhound coach the rest of the way thru to Broome.
May I suggest starting the journey by taking the XPT (train) from Sydney to Brisbane. Here’s the link to the NSWTrainlink Brisbane to Sydney timetable. I’d suggest getting a sleeper.
From Brisbane join the Spirit of Queensland train to Townsville. Here’s more info and timetables
From Townsville, you’ll head west to Mt. Isa. You can do this on the train called the Inlander. More information on the Inlander can be found here.
Once you’ve arrived in Mt. Isa it’s Greyhound coaches through to Alice Springs, Darwin and Broome.
Broome to Perth
From Broome, there’s a coach service to Perth provided by Integrity Coachlines. Full details and fares are available at their website.
Perth – Adelaide
The only train service from Perth to Adelaide is the Indian Pacific. There are NO coach services unfortunealey. The one way fare from Perth to Adelaide starts at $1,279. Check their website at https://journeybeyondrail.com.au/journeys/indian-pacific/perth-to-adelaide/
Adelaide to Melbourne
There’s the Overland train from Adelaide to Melbourne. Full details here.
There’s also a coach service from Adelaide to Melbourne provided by VLine
Melbourne to Sydney
The Melbourne to Sydney XPT (train) runs 7 days a week. Further info from the TransportNSW website