Get a few shots of Federation Square. It'll look a bit odd to you at first, but the lines grow on you.
You'll have to get a photo from the top of Swanston St looking down, through the heart of the city to the Shrine of Remembrance in the far distance.
The newly rebuilt Southern Cross Station is spectacular.
You should be able to get some decent photos of yourself with the Arts Centre spire in the background, but you'll need to be in the Alexandra Gardens to be far enough away to get it all in the shot.
There's something particularly Melbourne about Pellegrini's in Bourke St. Get a shot of yourself having coffee.
Nothing says Melbourne like the Flinders St Station clocks. They're as iconic as you get.
Sit on the big marble coin purse out the front of the old GPO building at the corner of Bourke and Elizabeth Streets.
Melbourne is slowly accepting the Docklands precinct to its collective bosom, so get some shots there if you get tired of the Southbank area.
I totally agree about the photo of the tram. If you can get a photo of one of the older model, W class, trams, with the doors that slide open, that's your best bet. If you can get a photo of one in one of the tree-lined streets of the city, like Collins St or St Kilda Rd, that's perfect.
Lastly, get a shot of the sun setting over the bay. You should be able to see this from St Kilda beach, but you may want to go to one of the beaches further down the Mornington Peninsula, even as close as Blackrock.