Basically nada. Not quite, though. A doctor's visit (pick your GP) costs $45 unless you are old, young or poor. Then it's a lot less. Free for kids 18 and under. Then, if you need tests, scans, surgery, whatever, all that is free. Also, once you have paid for 20 prescriptions, the rest are free for the year.
So that is the standard and that is what most Kiwis do. Being Yanks, we buy additional health insurance. We basically buy a low level of coverage, which runs us about $3000 a year for the two of us. It lets us skip the queue if we need surgery, and have it done at the private hospital instead of the public one. The deductible on surgery at that level is pretty substantial, but if you are willing to wait some and have it in the public hospital, it's free. Moms Skillethead just had cataract surgery and we paid about $300 to have it done right away.
I think the idea that you pick your GP, pay a moderate amount per visit unless you are young or poor, and then can buy insurance for premium care if you care to spend it, would make a solid approach for the US. Also, we cannot sue for medical misadventure. Most doctors are employed by the government and don't have to buy insurance. They can be put out of practice, but only very rarely sued.
We should probably put more money into our health system than we do (Kiwis tend to be a bit on the cheap side), but the structure of the system is solid.