and welcome aboard!!!
I live very close to the corner between Moreland and Melville Rd, and have been commuting to the CBD for 3 years nearly, so I know the area fairly well.
I luckily have showers at work so I couldn't help you with the suggestion you were seeking in your second post, but I can tell you that if you don't intend to ride at full bore you can probably get away without a full shower. Depending on how much you sweat, you will be probably be ok with towels, wipes, deodorant, and a set of fresh clothes.
As per the route...
1) My preferred (and fastest) is: cnr Moreland & Melville - down Melville - turn left on Victoria St - turn right on Sydney Rd - then straight ahead alongside Princess Park - across the large roundabout (2nd exit) and then straight on to QV Market. Only con is the amount of traffic, particularly on Sydney Rd. As suggested by another forumite above, it's a clearway so the bike lane is usually safe place to be.
2) The safest way is: cnr Moreland & Melville - down Melville - turn right on Albion, past the bridge on the Citylink, past the bridge on the Moonee Ponds Creek, turn left on the bike path, which you can follow until you get to where Flemington Bridge Railway Station is, then follow Royal Parade until the roundabout (2nd exit) and then straight on to QV Market.
I steer clear from the Upfield bike path (running parallel to Sydney Road) because I personally find it even less safe than Sydney Road: too narrow, very crowded, and usually the riders using it tend to be too slow for my pace. Nothing wrong with that, in fact it's me who would cause a problem: I tend to "race" to work (I usually average well above 30 Km/h), which, particularly in peak hour, could pose a danger to other path users
who clearly have not been bitten by the speed bug as I (unfortunately) have.
If you want to give it a try, I am happy to go together - just let me know (happy to take it easy on the cadence for the occasion - unless you're a speedster, which will force me to drag behind you!)