Welcome!
You probably know this but:
Get any gear you have fully checked out by a tech at a dive shop - especially life support equipment - regulators, tanks, gauges, etc
You don't have to buy new equipment; you can rent what you need and buy later - its a better way to go, believe me.
You might find that if the gear you "inherited" is old, out of date - things change frequently in the industry, mostly for the better - you may want to start fresh. Good eq makes diving SO much easier, for every diver, but especially when you are getting started.
While Melbourne is not a dive mecca, you are close to some of the best diving anywhere in the USA - from Palm Beach to the Keys, its great.
Take your time, get certified, rent equipment, talk to other divers, and read the board and you will be on your way - if the eq you have is good, fantastic - but my bet is - unless you are real lucky, or the person recently bought new eq - you will find you may want to replace much of it. Good Luck!