Welcome to commuting by bike. I have been riding a bike for over a year now to work and wouldn't go near Ballarat rd! Good work for trying. I would try all the suggestions others have given and see which one you are most comfortable with distance wise/traffic wise etc. From West Footscray I ride down summerhill rd from Barkly st to get to Highpoint (actually doing it tonight), but there would be a more direct route taking the quiet streets for you from sunshine.
Riding to the CBD, I would try to get on the bike path that starts at the corner of Ashley and Rupert Streets near Tottenham station. Then from there are many options to take to the cbd. I follow the path to the end, then go under the Geelong Rd bridge, down buckingham street, turn right on to victoria street, left at pilgrim st, left at Bristow, left at hyde, right at parker, cross whitehall st/docklands hwy (can get dangerous, but i don't mind this intersection as long as you are patient and don't risk your life - it is really in your control when you go) then onto the bike path at moreland street. Follow this path and it takes you to the CBD I like this way as it takes me via the back streets of seddon and the only dealings i have with trucks are at whitehall st/parker st intersection and when you cross is up to you, I have the control
I also only have one traffic light this way too and every street has a bike lane