I'm planning a four week trip to Australia in 2017. The main places I plan to visit are Melbourne, Sydney and the Great Barrier Reef. Any advice on the best time of year to visit all three?

I understand it gets very busy around August/September and over the Xmas/New Years period (Nov-Jan). Busy means accommodation is harder to find and more expensive, same with domestic flights, tourist activities, etc. It depends, as it does in Canada, with when the kids are off school and when stat holidays are.
We just got back from a month long trip a week ago (Dec/Jan) and did Melbourne and the Great Ocean Road, Northern Territory (Kakadu/Katherine) and drove in a camper from Sydney to Port Douglas/Cape Tribulation to see the Reef. It's summer down there right now and it's HOT! But it's also wet season this time of year and low season once the kids are back in school in January. I think it was a good time to go- prices are better, the weather was still great since we were just on the cusp of the beginning of wet season. Any later and it could have sucked a bit, rain wise- especially in Northern Territory which experiences some pretty incredible flooding in the wet season, but you're not heading there anyway.
I've been in July as well and it was lovely everywhere north of Sydney. A little cooler but once you're in Far North Queensland, it doesn't really matter anymore. Melbourne I'm not sure I'd do in their winter but I've never done it in the winter so hard to say. I imagine it would be much like a Vancouver winter- a lot of Aussies head North to Queensland during the winter. Depends on what you're ok with weather wise. I prefer to avoid rain. You'll get beautiful weather in all 3 in the summer, if you can handle the heat and humidity. (30-40C some days but not all the time)

Re: Melbourne winter - It is cold, like Vancouver. But it is still okay to do stuff outside as there are cold but still decent days in comparison to Vancouver's wet inside days. That said, it is definitely more lovely in the summer and there is so much more stuff going on.

I went on the exact same itinerary (Great Barrier Reef, Melbourne, Sydney) for the first three weeks of November. It was great because it as just the start of summer so not as hot as it could be though there were still plenty of 30+ days and even into 40 in Sydney one day. If going at this time I would suggest starting up north first at the Great Barrier Reef and then moving on to Sydney and finishing with Melbourne. That way it's not too hot yet up north and it will get warmer as you make your way down to Sydney and Melbourne because it will be late spring/early summer.

Yeah that's true. We had to wear stinger suits in Dec/Jan but not when I went in July. It is a bit of a bummer but your chances of getting stung are really low, just higher in stinger season. One guide explained that it should be viewed as walking through a rainforest. The chance of being bitten by a snake is very low, but higher in "snake season".

Thanks everyone
I was hoping that Feb-Mar would be okay since I can't leave work during January

It will be fine but expect rain up North, it's well into wet season by then. Always a chance of a tropical storm and flooding. It'll still be hot in Queensland though