Healthcare in the U.S for foreign graduate students
Unfortunately the way you access to health care will be largely governed by the specifics of your situation and the institution which you are going to, so we can't answer your question with any degree of certainty. The general answer to this kind of question is that you will probably have access to good health care but you might pay more for it than you would at home. Make sure that you have some savings that you can rely on in the event of unexpected medical bills.


It is common for full time graduate students to be offered some kind of subsidized health insurance through their university, or even require graduate students to participate in a group health plan
The affordability of such plans (how much you pay) can vary a lot by institution
It is common that such plans have restrictions on how you access care, for example it might require that non-emergency care is provided at the campus health clinic
Since the passage of the Affordable Care Act (aka Obamacare) your health plan providers are generally required to cover any common illness or treatment. However, there are current political efforts to repeal or defund the ACA, so this may change in the future. In particular the ACA greatly expanded coverage specific to women, pregnant women, and children.


You should direct questions regarding health coverage to your university's international student's affairs office, as this will no doubt be a common question for them. They will be able to tell you how their graduate students access health insurance. Many existing students are more than happy to talk about their experiences as well- ask your future department whether there is a graduate student or international student group that would be willing to answer questions.

Edit: Update on how to plan for medical expenses. Yes, unfortunately it is common for major medical events to cost several thousand dollars in the US. Your friend paying $2000 for a broken arm is entirely possible.

The coverage of health insurance in the US is commonly expressed as three numbers: the deductible, the out-of-pocket maximum, and the coinsurance rate. These can give you a ballpark estimate for how much health care might cost you.


Deductible: How much you must pay, in a year, before your health insurance pays anything
Out-of-pocket maximum: How much you must pay, in a year, before your health insurance covers everything
Coinsurance rate: How much your insurance pays after you hit your deductible, but before you hit the maximum.


For example, my grad student plan was:


Deductible: $350
OOP Maximum: $7,150
Coinsurance rate: 80%


Suppose you break your arm and go to the hospital. The hospital sends your insurance a bill for $5,000.

First, you are responsible for paying up to your deductible without any assistance, so you pay the first $350.

After you have paid the deductible, your insurance kicks in and pays 80% of the bill. The remainder after deductible is $4,650 and your insurance will pay 80% of that, and you will pay 20% of that amount.

Your total cost would be $350 + $4,650*20% = $1,280. 

Suppose later in the year you get sick and go to see the doctor. The doctor bills your insurance $100. Since you've already met your deductible for the year your insurance pays 80% of that $100 and you pay 20% ($20) for the visit. 

Your deductible and out-of-pocket expense resets every calendar year (January 1st). If you went back to the doctor on January 2nd and they billed you for another $100 then you'd have to pay that entire $100 on your own, and you'd have $250 left on your deductible for that year.

So, how do you plan for medical expenses? If you have a medical crisis then you should assume you will pay at least as much as your deductible. If your deductible is $500 then have at least that much in savings (you should have 6-9 months of living expenses in savings anyway as a precaution).  If you have a really serious health problem, then I would assume that you're going to pay up to your out-of-pocket maximum, and that is your worst-case expense every year. Additionally, if you have a chronic health problem that requires a lot of management, it might be wise to assume you're going to pay your out-of-pocket maximum every year.

It is not uncommon for people in the US to go into debt due to a medical crisis. A lot of people can absorb a $500 medical bill, but not a lot of graduate students can absorb a $5000 medical bill. If you cannot pay your bill in full within a month or two, then the hospital will start charging interest as though it were a loan. 

In a personal example, I had a health crisis about a year and a half ago that required a two week stay in the hospital. The hospital billed my insurance several hundred thousand dollars. I paid my out-of-pocket maximum out of savings, after which I am no longer responsible for the balance. After haggling back and forth the insurance finally paid about $150,000 to the hospital. So yes, health care in the US can be obscenely expensive. 

Note that a lot of health insurance plans will cover routine care for free. Annual physical exams were paid at 100% by my insurance provider, and separate annual gynecological exams were covered at 100% for women. Same thing for routine screenings (prostate, mammograms, etc.) and immunizations. This is because preventative care is cheaper in the long run than treating acute or chronic conditions that could have been prevented.

Again- find graduate students in your prospective school who are willing to talk about their past healthcare experiences and how much it cost them. They will give you a reasonable expectation.