Love your idea and love the idea of giving you ideas!!
I'm a transplanted Aussie with the advantage that I've visited every area of Australia so I'm adding my thoughts as well.
For starters, check out the visa requirements.
Then I would suggest Brisbane which has a beautiful botanical garden on the Brisbane River. It's a mecca for backpackers (among other tourists) so accommodations can be found somewhat cheaply. Since it's semi-tropical, it's quite warm in the Australian summer. It's not right by the ocean but you can drive there - or catch a bus or train, particularly if you're going to Surfers Paradise. Public transportation all around Australia is quite unlike the U.S. so travelling by train or bus isn't a problem.
I was also going to suggest Darwin which has a great botanical garden and a good bus service. It's on the ocean, except that you can't always swim there. And there are crocodiles in the waters. Not good! Maybe give that one a miss.
I agree with Perth or Fremantle too. The only drawback is that it's a long trip getting there - but well worth it when you arrive. Lots of gardens in Perth, botanical and otherwise so do check it out. Fremantle might be even better. Beaches, gardens, artsy, etc.
Difficult to find Australian cities with botanical gardens which are also on the ocean. Of course, there is always Sydney where I'm from. Gorgeous Botanic Gardens right in the city, well on the Harbour front. Beaches are out a way since the city is on Sydney Harbour. But a good bus service can get you anywhere you want to go to access the northern beaches and a train service takes you to famous Bondi Beach. But, Manly (one of the oceanside suburbs with lovely beaches and accessible to the city by ferry and hydrofoil), has gardens, and may fit the bill for artsy besides. And there is always Oxford Street which has lots of artsy stores and the suburb of Newtown, quite close in to the city is very artsy. Lots to recommend Sydney too, I guess, except its size and busyness.
So I'm still leaning in favour of Brisbane, and Fremantle.
Anyway, I love your idea and it's been fun to think about just the right place for you to while away the snowy days others will be experiencing here.