I don't know about the US system but work in and know the Australian system. If you are planning on going private, I wouldn't even bother with the public system, you will be triaged down the list and could wait ages just for an appointment and then longer on the surgery wait list.

The best way is to get your GP to write you a referral to a general surgeon in your area - GPs can be wonderful but it's always good to ask the expert about it, have all your questions answered. Most general surgeons don't have long wait lists to have an appointment, whereas I find that even private gastroenterologists can have long wait lists. And you don't have to commit to having surgery just because you see a surgeon. The appointment will cost you the gap that medicare won't cover - usually about $100 but can be very variable. And if you decide to have a procedure done you will also be able to choose a convenient time around work and kids etc.

Hope this helps, the medical system can be hard to navigate and even though I work in it, it can get confusing and the administrative details change all the time. GPs have the worst time having to constantly be up to date with the government changing medicare rules all the time.