No, American does not fly to Oz. The airlines that fly to Australia are United, Delta, Qantas and V Australia.
Tickets have been cheaper than ever this summer with fares on Qantas, United and Delta all as low as $700 USD from JFK(via LAX or SFO) and $620 from the west coast. Don't expect fares to stay that low for long though cause winter is ending down under and the fares will rise. In general, try and avoid United on that route at all costs as they fly ancient planes without PTV's. Qantas is an excellent airlines and Delta has been flying their newest planes on the LAX-SYD route. Expect to pay $1000+ in the winter but try and see if you can find better deals because of all of the new competition on the route now that Delta and V Australia have begun flying to Oz.
As far as tips on Australia, please give me some more details on what time of year you are traveling and what sort of things you'd like to do and see. Generally, I would recommend some combination of Sydney, The Great Barrier Reef & Gold Coast and a few days in the South Island of New Zealand if you have the time. Melbourne is nothing to fuss about and Perth is far, far away! PM me if you need more info, I'll be happy to help you out.