Hi Jerzy,
I've been diving in Melbourne for around 15 years, so I figure I'm qualified to answer your question
In December, you'll likely be getting outside temperature anywhere between 25ºC and 40ºC (77ºF to 104ºF) Water temperature will be around 18ºC (65ºF), depending on whereabouts and how deep you go. A 5mm may be OK for you. I dive a 7mm semidry 12 months a year in water temp down to 10ºC (50ºF) in winter.
The Melbourne CBD is at the most inland part of Port Phillip Bay, and there's bugger-all diving around there. To get to the real dive sites, you need to head down the coast in either direction (usually South East, towards Portsea) about 1½-2hrs drive.
The most popular dive operator is
Dive Victora who run three boats (I think) out of Portsea, and usually half a dozen dives per day. I usually dive with
A.B. Ocean Divers who are more club-based, and run their boat usually every Sunday and local shore dives on Saturdays.
Expect to pay around AU$50 for a dive, plus gear rental.
if you want shore dives, there are a few good spots in the bay closer to the city, but depths are shallow, and usually in the 4m-8m range (12-24ft). Good spots are Brighton, Black Rock and Beaumaris (about 20mins from the city) and a then a bit further out Mornington Pier (which is about half way between the city and the dive operators at Portsea. For most of these you will probably need a local diver to show you where the reef is. Most non-diver Melbournians don't believe me when I tell them there is more than just sand at those spots!
You'll see plenty of sting rays, nudibranchs, snapper, banjo sharks, old wives and boarfish no matter where you go. If you dive off the boats, some of the other sights are sea horses, leaft sea dragons, seals, port jackson sharks, cowfish, blue devils and if you look hard enough a few crayfish.
If you decide to get wet, some of the highly recommended boat dive sites include the J4 Sub (also called the 27m sub), anchor wall, boarfish reef and castle rock. You'll definately also want to get onto one of the drift dives. A whole lot of water flows in and out of the Port Phillip Heads, which are only just over 1km wide, which can make for some serious drift current!
If you want warmer water, you're going to have to go a long way out of Melbourne. If you're flying, Sydney is the first stop. If you're driving, probably somewhere like Eden or Merimbula, which are about 5hrs drive away, and the water will be no more than two or three degrees warmer.
My recommendation is get out on a dive. Hire a 7mm and give it a go - they're not that different to a 5mm!! If you head down to Portsea, the shop is right at the pier, so you can put the kit on, and go for a try out dive in about 5m (15ft) under Portsea pier to test the water, wetsuit and weighting.