From what I've heard, avoid TVSA.
You can easily do flight training with many of the schools at Moorabbin. I personally train at Tristar Aviation, but there are many options around and I would urge you to have a look around. I'd imagine most people training with recreational purposes in mind tend to fly part-time, about one hour a week. I'm working at this rate, and fitting it into Uni study quite nicely. You'll just need to either have the self-discipline to self-study the theory at home or arrange your time such that you can attend ground school for the theory components.
Here's some links to get you started (a lot of these sites are horribly designed, but yeah):
http://www.rvac.com.au/ - Royal Victorian Aero Club
http://mfs.com.au/ - Moorabbin Flying Services
http://tristaraviation.com.au/ - Tristar Aviation
http://melbourneflighttraining.com.au - Melbourne Flight Training
Unfortunately, flight training
is an expensive hobby and it is not really something that can be subsided. Basically, it's around $200-300/hr (depending on the type of aircraft and who you fly with), and you'll need to have around 20-30 hours for your General Flight Progress Test (GFPT - allows you to take passengers around Moorabbin) and a similar amount for your Private Pilots Licence (PPL). To get your PPL, you're looking at around $10-15k depending on how fast you learn. Furthermore, the additional costs for gear like headsets and the costs that the Civil Aviation Safety Authority (CASA) slug you for your ASIC/AVID, SPL and Medical adds to the pain. I think I've spent nearly $4k already!
The only subsidy available really is to start off with the intention of completing a Commercial Pilots Licence, which is GST-exempt (so, effectively, 10% saving). Even then, you'll probably find that a good number of pilots are probably living below the poverty line...
You can join the Defence Force as a pilot, but I don't think they do part-timers. They also lock you in for 12 years or something, so it's probably not something for you.
@ dekoyl - I don't have that many hours! 12 hours is not that much!