It's Machka, actually.
Getting out of Tullamarine Airport is not that difficult by bike. Turn left at the traffic lights if you are heading west or northward. Even the ride through the burbs to the Melbourne city centre isn't too bad.
The Western Highway is the main route between Melbourne and Adelaide, but while the shoulders are OK (littered with debris and narrow over some of the bridges) you will need to be ultra-tolerant of fast-moving heavy transport and even fast-moving car traffic.
To get to the Western Highway, turn left out of the airport at the traffic lights, and continue on the remainder of the Tullamarine Highway (after 500 metres it become the Sunbury Rd), then continue much further until the turn left at the junction for Diggers Rest. At Diggers Rest, cross the Calder Highway, and continue on the Diggers Rest-Colmadal Rd to the Gisbourne Rd where you turn left towards Bacchus Marsh. You'll join the Western Freeway before you actually get into Bacchus Marsh. You'll have some... interesting... climbs out of there.
The Western Highway continues on past Ballarat, and north-west of there, you have the option of continuing on the Western Highway, or turning right and northward on the Sunraysia Highway. The latter should be quieter, and you can head north until either the Borung Highway that will take you back on to a (hopefully) quieter Western Highway much further to the west; or you can keep going until Ouyen, then turn left on to the Mallee Highway to Adelaide.
I notice that your original intention was to go through to Mildura, which is entirely possible. I haven't been on the highway between Mildura and Adelaide (I likely will do this fruit-picking season), but as I just said, this is fruit country with lots of orchards, so things aren't quite so desolate.
to see what I am talking about with the routes.
You could possibly take the Calder Highway, but it is a more circuitous route to Ouyen, and it gets a bit hairy.
Personally, I would take the coastal route to Mt Gambier, but I also like riding up through the mid-west towards Ouyen and on to Mildura. The Mallee Country has something really appealing about it, but it's not for everyone.
Anyway, the western and mid-north-west sections of Victoria have been through some uber-drought. Expect there to be long stretches of straight, flat, seemingly boring roads. The winds are predominantly from the north-west through west to south-west. You will need to pack enough water for up to six hours riding -- the "towns" marked on the map may be single-house settlements with no services.
The people through here are good, honest country folk who would do anything for you. I am not sure they will identify with an English charity they've never heard of, though, so don't expect to raise too much on the way from them.
I think north of Port Augusta, things change a little bit. Port Augusta 10 years was my least favourite town when I did my Perth-Adelaide crossing. The roadhouses are oasese, and you should not have to send ahead food packages for any part of your trip. You can do it -- I think Australia Post has send-and-hold services. But really, you would be spending as much as it would cost to buy food along the way, even at inflated prices. You might have offbeat eating habits, however, and frankly, vegetarian is not someting they cater to that well. You can still survive, though.
Your trip from Adelaide to Darwin is not unique, and there are others who have done it. The most recent journal I have read is on www.crazyguyonabike.com
, and was done by an American. He had no trouble finding what he needed the roadhouses along the way, even if it was not always to his standard. He pulled a BoB trailer for his gear. An English friend did the trip two years ago, too, and his general comment was: Watch out for the roadtrains -- get off the road, because they don't move aside, and the dust-storm they leave in their wake is pretty thick.
I know that headwinds on the south-north route can be an issue in Central Australia. Certainly, make sure you have enough water capacity to sustain you for several DAYS. And it can get mighty cold at night in Central Australia despite its heat-and-desert reputation -- lows of 0 deg C aren't that unusual. Plus you'll need to pick the timing of your trip around the northern monsoonal season. If that bulldust gets wet, it will stick like pooh to a blanket and when it dries, it is like cement. Don't go off the main highway unless it is obvious that your destination is well marked and well trafficked.
I also trust that this is not just a whim thing, and that you do actually have some experience in extended cycle touring, and that your fund-raising is for a reputable organisation. Check the Touring forum here for some sagely advice if you don't have much experience.