You would do the same thing you do today. There's a few options, you could go to your primary care physician who would have a known price for an office visit. You could go to an urgent care type facility. You could go to something like a CVS Minute Clinic which has flat rates for "minor illness" visits. All of whom may discover you have something more serious and refer you elsewhere.
You are acting as if paying out of pocket at a price not determined by an authoritarian central planner would suddenly mean nothing in healthcare would be regulated in any way and people would be getting healthcare from crackhead joe in the back alley.
Not everyone will pick the cheapest option all the time. Today there are free clinics in many places. Most people don't go to those. Some people will choose Wal-mart. Some people will choose Macy's. And there will be a myriad of options in between.