Will be in New York in September from Australia - is there a website of somewhere I can find out what shows will be playing on Broadway and is it adviseable to buy tickets before we leave home or would we have no problems buying them there? Does New York have cheap tickets for unfilled seats on same day performances like London? Any show suggestions?

It may be advisable to buy tickets before you leave Australia, depending on which shows you plan to see. For some show it is very difficult to get tickets on short notice. Which shows did you want to see.
There is a ticket booth in Times Square that sells discounted tickets on the day of the performance.

Look on playbill.com This website lists what's playing & also offers discounts for certain shows. You have to register (like here) for free to see discounts. There are lots of websites like that including

hitshowclub.com
theatermania.com

There's also an "offical" NY theater website but don't have it off hand.

If you want to see a particular show that's new or very popular, it' better to pay full price (or more) & buy ahead of time. But if you just want to see a show, you can either get discount tkts at the Times Square booth (there's also one at the South Street Seaport) OR use one of the websites.

Current "hits" that have been playing a while are Hairspray and the Producers. These are never available discounted but by the time you come, that may change. Wicked, Never Gonna Dance & Wonderful Town are newer musicals. Teh Tony awards (equivalent to the Oscars but for Broadway) are in March. That will make the winners harder tickets.

The Tony awards are always the first Sunday in June -- not in March. But considering your great and complete answers, mclaurie, I think we can forgive this one little "mistake".

As the previous poster said, whether you buy tickets in advance or wait until you get to New York really depends on which show you want to see. Early September is a fairly easy time to get tickets. Later in the month is harder. Do you have any preferences for shows? Of course, coming from Australia, you might want to see The Boy from Oz, since it's about Australian Peter Allen and stars another Aussie, Hugh Jackman.
One correction on mclaurie's posting, the Tony Awards are given out in early June, not March.

A couple of years ago I was walking up Seventh Avenue through Times Square on the Sunday of the Tony's. I ran right into Richard Dryfuss (who I believe hosted) and about knocked him off the sidewalk. Along comes Nathan Lane and Bette Midler. I wonder what the heck is going on and realize it is the Tonys. Not much of a red carpet but they were held at Radio City.

KA, buy tickets for the more popular shows. I would not wait much later than May or June to purchase tickets ahead of time for the more popular shows. TKTS is always advisable for same day.

September is one of the busiest months of the year so shore up your accommodations.

Patrick is the board theater guru so any advice he gives will be excellent.

Sorry folks, don't know what I was thinking. Must have had the Oscars on my mind but now I understand they're on Feb 29! Forgot this was a leap year til I saw that.

That's okay mclaurie, you're still okay!
Also, not to diminish Patrick's reputation, but Bugswife is definitely the definitive New York theater expert. She sees everything--and I do mean everything--and often two or three times. (Hey, Patrick, I consider myself a raw recruit next to her!)

Thanks everyone for the feedback - New York is so spoiled for choice its fantastic compared to the one major show we are lucky to get a year. Anyway had a look at some websites and thanks HowardR forgot our Hugh Grant was doing Boy from Oz which is an option, also considering 42nd street, Hairspray based on your comments, and the Lion King any preference or better alternatives?

The number one show for you to see -- Hairspray.

Just realised that I must be having a complete brain fade today when I said that Hugh Grant was in the Boy from Oz -obviously I meant to say Jackman!

Thanks for the tip on Hairspray Patrick.

Just saw Wicked....was terrific!

Oh, I'm so glad you clarified that, AK. When you said "our Hugh Grant" I somehow thought you meant that you consider Hugh Jackman to be the Australian counterpart of Hugh Grant. I didn't get the comparison, so glad that's NOT what you meant.

Thanks Doug for the feedback to Wicked -I think I have pretty much made up my mind to see Hairspray (this is probably what I'll start with anyway!!! who knows how many more shows I might have to squeeze in while I'm there). Thanks Patrick for making me feel good - was hoping nobody would notice the slip up but then thought better of it to admit to the error and like you said you thought that we must have some strange attraction to Hugh Grant here in Australia which I can safely say that we don't!