I'm in Australia myself this March/April but sadly only for 2.5 weeks so very jealous of the OP having a full month. I've been quite a few times as I used to live in Asia and one of my good friends lives there.
I love Melbourne!! (and would happily live there) If you're into coffee, there's a great half day coffee tour or just make sure you stop for coffee at Patricia and/or Captains of Industry. It's definitely worth a drive along the Great Ocean Road (from Melbourne). You can join a small group day tour (long day but it shows you the highlights and you don't have to do any planning yourself) or hire a car and do the drive yourself over 2 - 3 days. Note: you will need to book places to stay ASAP as leaving it until 2 months before is quite late so a lot of places will have been booked up months ago. The first time I went to the GOR it blew my mind, it's just stunningly beautiful! It was also the first place I saw kangaroos and koalas in the wild. It's still one of my most favourite trips. If you're into animals/wildlife, a trip to the Penguin Parade at Philip Island is a must-do. It sounds horribly touristy (I hate tourist stuff!) but it's so well organised and well done and just incredible seeing the little penguins so close up. If you go, do the ranger tour as it's definitely worth the extra money. I go every time I'm in Melbourne and my friends all laugh at me, then decide they'll come too (despite being Melbourne locals).
If you like, there's great wine tasting trips to be done around Melbourne and Sydney. I really enjoyed the Yarra Valley, perhaps because it was my first Aussie wine tasting experience, and I also really enjoy the Mornington Peninsula as it's a bit more local rather than geared up for your groups.
In Sydney, you need to do all the cliches like get the ferry to Manly, visit the Opera House, go to the Blue Mountains etc. I much prefer Melbourne to Sydney but you might be the opposite OP.
And to throw in a curveball or two, have you considered a side trip to Tasmania or Uluru (or both)? Tasmania is my most favourite place in Aus, it's just incredible. And the foodie scene in Hobart is very hot right now (my Sydney friend is actually coming to Hobart with me for this reason!).
I found Uluru just stunning and quite moving actually. I really didn't want to go (aforementioned hatred of tourist stuff) and when we were landing, I could see the rock out of the plane window and was "that's enough for me now, we don't need to land, I'm done". But when spending time there (2 nights is about right), it really surprised me as I enjoyed it so much, and it wasn't quite the tourist hell I was expecting. Drinking a glass of champagne while watching the sun set over Uluru is pretty special.
If you think you might end up in Tasmania at the end of March, I'll be in Bicheno drinking wine and watching the little penguins come in to the burrows in the evening and you're welcome to join me for a glass or two.