Australia is a big country. It's a little like ... , well, actually, it's a *lot* like, if not *exactly* like saying, "I'm going to Canada. What should I see?"
Decide on what parts of the country you want to see, no way you're going to do all that you want to do in 3 weeks. Budget accordingly. Flying from Sydney to, say, Cairns will set you back several hundred dollars per seat. It's a 3 hour flight IIRC.
Sydney is a really neat city. The harbor is amazing. Sydney is worth a few days stay. This is probably where you will have the best "beach" time this time of year. Watch out for the great white sharks though
The Sydney Aquarium is really well done and worth a see IMHO.
Queen Elizabeth park is one of the most amazing gardens I've ever visited. Watch out for the flying foxes.
They roost in the trees there during the day. They number in the thousands if not millions. At night they fly across the harbor and fly in the spotlights illuminating the Harbor Bridge.
But to be honest, expect Sydney to suck a lot of your funds. When I stayed in Sydney (my sisters family lives near Sydney, they used to live IN Sydney but not anymore - so we had to hotel it). And yeah, um.. $300 a night. It was a really cool condo right on the water across from Luna Park, but yikes that was a pretty penny.
The reef is a must see, but bear in mind that you cannot just walk off the beach and expect to see reef. In fact, this time of year, you can't go to the beach in Queensland, the box jellies are in season. Plus, the water is poo brown for several kilometres off shore. Don't plan on any beach time if you go to Queensland. Well, you can sit on the beach if nothing else, and I guess technically you can still go for a swim and just hope that the jelly nets hold. But they do get holes. And technically they can drift down south quite a ways. My nephew was stung by a box jelly near Ulladulla (NSW, south of Sydney). It looked like 3rd degree burns starting from his neck/shoulder, through his torso down to almost his knee. Pretty nasty stuff.
Anyhow my point being, if you want to see the reef, you have to go on a boat and they will take you out on a daytrip. Or do a liveaboard, if your budget allows, do a 2-3 day trip out to the reef.
Any real reef you might have seen, is chump change once you see the GBR and other South Pacific reefs. In terms of reefing in the South Pacific, Papua New Guinea ("PNG") is roughly "ground zero", the further you get away from it, the less diverse the reef gets. So PNG, Indonesia, Australia, etc. reefs are really the pinnacle.
I myself would not go to Australia and not go plan on spending at least a week in North Queensland. When I was last there, I stayed between Cairns and Port Douglas, and actually planned my reef excursions out of Port Douglas. However, I'm told that nowadays Port Douglas has become quite trendy among the rich set and the famous of Australia and so has become priced accordingly. Think Banff but worse.
Soooo ... I suggest looking into it, but if you indeed find it a bit too pricey, look at Cairns (you'd be flying in and out of Cairns anyhow), or Townsville.
Townsville also has an aquarium, the Great Barrier Reef Aquarium. It's probably worth seeing although I haven't seen it myself. But it's referenced in a couple of my reefing books (can't remember if I saw it in Fossa and Nilsen, or Delbeek and Sprung, but those kinds of reefing books).
What else can I tell you ?? Man, I've already written a novel.
Other things to consider seeing: Kangaroo Island, Fraser Island, Heron Island, if your budget allows. These are all on my "places I have to go see one day" list. Uluru (formerly known as "Ayers Rock"), The Twelve Apostles ... also neat to see, but probably unrealistic to get to if you plan on spending any time already in Sydney and Queensland.
Adelaide and Brisbane areas also offer spectacular beaches. And spectacular surfing, if you're into surfing.
Melbourne is a pretty neat city. Like Adelaide though, these cities are sort of on the wrong side of Sydney if you want to go north from Sydney. You can do it but it depends on how rushed you want to be.
Hope I've given you some fodder for thought.
Enjoy your stay, it will be great.