Is graduate school free?

I hear that people get paid to be TA’s, but I also know that people take out loans to pay for graduate school? How much does graduate school end up costing after all this and how do people usually pay for graduate school (to get their PhD’s… not masters). Also, my parents are paying for my undergrad, so I won’t have loans after my BS. They probably wont (and shouldnt) be paying for my graduate school (if I ever choose to go) so would I take out loans? How do most people pay for graduate schools? I know people take out 300k loans for med school… is it a similar case for graduate school?

PhD programs are not worth doing unless you are funded by the department. If you live frugally, you can manage easily on a TA or RA stipend and the university will pay your tuition. You can even have savings at the end of your degree. I suspect that there are some programs which do not provide support for PhD students but they are certainly not in STEM fields.

How often are people funded by departments if they plan to get a PhD in math? How do people increase their chances of getting funding? Even after funding, how much do students end up paying for graduate school?

“How often are people funded by departments if they plan to get a PhD in math?”

Depends where and for whom. At UC Berkeley many departments only have PhDs that are fully funded. But getting in is a highly selective process.
DS is in a fully funded graduate school in France. But then, Europeans are different when it comes to higher education. He has friends who moved on to full-pay (and it is expensive! But they have healthy trust-funds) graduate programs, friends who have partial fellowships, and friends who got into fully funded graduate programs. So it depends.
It would be good to see reliable stats on this, I agree. Maybe some of the mavens of such sources will chime in.

In the U.S., math PhDs at reputable departments are virtually always funded. Generally speaking, if the department cannot fund you they don’t admit you. This is why the process is highly selective. In return for the funding, you usually perform some tasks - serving as a TA at least some semesters. In most science departments you would also serve as a lab RA, but I’m not sure what RA work would entail for a more theoretical subject like math.

At most schools the funding will be somewhere between $25,000 to $35,000 per year plus tuition coverage and health insurance. This should be enough to live on frugally for a few years IF you are only supporting yourself. If you have to support a nonworking spouse, children, or other family members then it’s not really enough. Those people often do take out loans to cover any shortfall. Sometimes they work a supplemental job - some departments don’t allow that, though. There also may be small things you borrow money for - like I borrowed money to cover the expenses of moving to a new city during my graduate degree, but that was a small amount.

A very few (and usually not reputable or not highly-ranked) departments may offer you admission without funding, but you should always decline this.

But no, you should never have to take out anywhere close to $300,000 or what medical school costs for a PhD.

As others have noted, most Mathematics PhD programs will support their students throughout. However, I would never say that getting a PhD is “free”. You have to consider the lost income form your 6-7 years earning just enough to manage as a graduate student. This is a decision not to be taken lightly and so you need to make sure you are doing it for a good reason, such as a passion for research or as a path to a career that requires a PhD.

If you are planning a professional master’s (e.g. MBA) then you will be full pay, less some merit aid at a few schools.