Has anyone stayed short-term in Australia (melbourne) and dealt with cell phone needs? Have Verizon which doesn't have a Sim card....is it best to buy (rent?) a local phone? Where? Thanks.

Depends on your phone needs. I would probably suggest you just buy a phone. U can buy a pretty decent handset and prepaid simcard quite cheaply(load credit as you go, tends to expire each month) from most of the big telco companies. Data (i.e. iphone, blackberry) is a bit more expensive.
The main companies here are
Telstra (which also co-owns the 3 network), Optus and Vodafone; or try one of the smaller companies - which include Dodo, Virgin, Boost. Check out their respective websites.
If u wait till u arrive, go to one of the company stores (they are all over Melbourne) Alternatively, you can go to a phone outlet e.g. stores such as Crazy Johns will stock phones from several different companies. Also, you can pick up a phone from most electrical retailers (try JB HiFi, Harvey Norman or Dick Smith) supermarkets or any of the department stores - e.g. Big W, Target, K-Mart (which are lower end stores) right up to Myer and David Jones (higher end). They will all have relatively cheap phones (I've seen basic models for less than $30AUD, credit can be as little as $15AUD a month).
Generally, phones here can be used in Europe and Asia - but not the US/Canada. So u might want to keep the handset for your next European vacation.

Honestly I'd recommend getting a cheap phone and SIM from Australia Post (our post office) - they have lots of phones at the lower end of the spectrum. Telstra charge premium rates for their service and so I'd only get a Telstra phone / SIM if I were travelling in areas outside the big cities, where they really excel. Optus and Vodaphone have patchy coverage in regional areas and I wouldn't rely on any of the smaller providers for country areas. However, if you're just in Melbourne and won't be travelling to the country, there are lots of cheaper SIM providers. If you have a phone that can be used here (a GSM will do it), Lebara Mobile have extremely cheap rates for phone calls overseas, if that's what you want your phone for.

Lavandula

Thanks so much. All of the above info really helps!

This might be of some help for Australia Post; you can't buy phones online but they have a catalogue:

http://catalogue.auspost.com.au/portal/


Lavandula

I have cheap phone that takes a SIM card which I use when we go to W.A. You can buy time at airports and just about any newsagent. For our minimal needs, we get Telestra for about $40 for six weeks.

You may want to have a look at Dick Smith's too, which is a wide-spread chain of electronics shops. Just for a variety of the brands listed above, in one place. When we moved to Melbourne I worked off an inexpensive handset and sim I got there. (Virgin. I'd probably pick a different brand if I were to do it again)