Great to hear you want to visit the great southern land of Australia. It’s a very large place, so lot’s of places to lose yourself in!
The Formula 1 race is quite epic, and you’ve picked a nice spot to land in. Melbourne is a bohemian paradise, filled to the brim with all sorts of cultures and not-so-subtly injected with a touch of hipster. I’d recommend checking out some of their laneways for politically-driven and socially-lauded art, as well as shopping. Also, if you like a drink or two, do some research on fancy underground bars to visit. And St Kilda is cafe heaven, as a side note.
As for climbing, you’ve picked some great alps to mount! Just be sure to pack some warm clothes. It starts snowing on the mountain around late May to early June (Australian winter is from June-August). I’d recommend to hang about til then and visit Perisher Ski Resort or Thredbo Ski Resort. You can hire equipment at a few stores in nearby Jindabyne.
If you like the idea of soaking up our colonial history, definitely head down to Hobart in Tasmania. The whole state is laden with historical sites since it was one of the earliest colonies when the British arrived (or invaded, depending on which side you’re on).
If you love trekking and climbing, I’d definitely recommend a visit to Cradle Mountain in the centre of Tasmania. The whole area around there is a pristine wilderness, under the protection of heritage society. I’ve written up a road trip guide which you can read here. There’s also Tasmania’s North Coast which is highly underrated.
When you’re in Hobart, visit the MONA as well. One of the quirkiest museums I’ve ever seen!
If you’re up to visit NSW and Sydney, I’d recommend during the warmer months. There are great beaches along the south coast of NSW (one of the best is Hyams Beach in Jervis Bay). Canberra is nice to visit for a weekend. Its top tourist attraction is Parliament House.
Darwin is the smallest of the capital cities but teeming with war history.
Perth is a secluded city on the other side of Australia. It has some wonderful beaches to visit, as well as a handful of tourist attractions. Highly recommend visiting Rottnest Island!
And finally, Adelaide. They like to think they were the only city to be built by free settlers, but history is catching up with them on that front. They had convicts around there too. That being said, Adelaide is the most efficient city to be built in Australia.
I do hope you enjoy your holiday here!