PhD student here. I am especially prone to stress, and my PhD has been very stressful, and while I haven't been completely successful with coping, I know a thing or two about it.
First off, I assume that you have gone through some schooling prior to grad school, which is what got you here (BSc, Masters?). That was stressful too, no? I have heard from SOME PhD students that undergrad was actually worse in the long run, but that depends heavily on the person, the lab, and the supervisor.
What you can do to stave off stress depends on your program and your supervisor. The healthy recommendation is to limit your working time. Some people on this site will tell you that a 'normal' working week is fine. Others will tell you that 60 or 70 hour work weeks are standard. This thread may devolve into a debate between those two camps. Personally, what I do is I set my own hours according to when I work decently. If I don't need to be in the lab for an experimental reason, I'll allow myself to get enough sleep that I'm not falling over myself through the day. If that means I get in a little late, it also means I stay a little late, especially if I get into a groove when working.
Sometimes, if I don't have stuff going on in the lab, I stay at home and I work all day in my pajamas wrapped up in a blanket, analysing data or writing a report of what I've done. This is especially true if my previous week was spent in the lab (I work in cryogenics and when we use liquid helium we have to pack as many experiments into 100l of lelium as possible, which means being at work from 8am to 9pm sometimes). I let my supervisor know about this, and I answer emails, and I do actually work (usually). But that means I can take breaks when I want to, be in my comfort zone, and relax a bit more than usual. I'm an introvert and it helps me recharge. If you're an extrovert, maybe it's not a great idea. Again, you do you. That's the important thing.
I try to keep weekends for myself, but this won't always be possible. The point is to try to give yourself SOME fun time, even if you are bogged down with deadlines. The best way to avoid a stress-related breakdown is to have a life outside of work. That means friends, hobbies, and other things unrelated to what you do. Even if they are other PhD students, it's still better than nothing at all.
Sometimes at night, I have trouble sleeping because the problems of the day keep running through my head, or I'm worried about something going on tomorrow. I installed a program on my computer called f.lux which automatically limits the blue light coming from your screen after sunset. This reportedly stops the limitation of melatonin production if you are working into the night, and can help with sleep if you've been in front of a computer all day. It may actually work, or it may be placebo, but I've been sleeping better. Also, if you switch it off after you've gotten used to it, your screen looks like some kind of crazy holy entity. More generally, practice sleep hygiene. It really does help.
Another thing to watch out for is seasonal affective disorder, if you are anywhere remotely northerly. I live in a not-very-north place (France) and I managed to get a case of it pretty bad this year. Keeping a desk near a window helps, as does walking somewhere in the middle of the day (e.g. to a cafeteria for lunch). Going for a run mid-day (if you can) helps too. Don't be surprised if you get the winter blues, especially if you wake up before sunrise and leave for work during it, and if you go home well after sunset at night. It happens.
I'd recommend living alone so you can rest and recharge at the end of the day, but if you're extroverted, this may actually be worse for you.
Finally, maintain a healthy relationship with your supervisor. If you find yourself worrying about whether you are progressing properly, talk to them about it. Be sure to maintain your independence (that's the point of a PhD) but also go to them for guidance when you need it.
Be sure to keep good records of everything you do (lab book if experimental, and make PDFs of your results written up nicely) to avoid stress later when you have to write a thesis. I experienced it in my masters; not having good records, and forgetting what you did before. Write it all up. Not only is it weirdly relaxing to write up reports, but it will help you later.
If you're in the US, you will probably have courses and qualifying exams. We don't have that here so I can't really comment, but the usual advice applies; start work early, create a decent schedule for yourself, don't leave a deadline until the last minute. Work with your cohort; they are a great source of support as long as they are not overly competitive.
And finally, the first few months of your PhD, I have been told, no one really expects you to actually accomplish much (YMMV with institution, program, country, and so on). Learn your way around, learn how equipment works (make great notes for this part!), deal with administrative stuff. Don't pressure yourself into being a magical science genius who spits out papers like an academic wizard. I spent a week translating an instruction manual for a dilution fridge from French into English. I spent a month filling in forms. Go to meetings, go to journal clubs. Part of the point is networking and expanding the way you 'think', so don't think of journal club or seminars as being a waste of time. It's partially a way to unwind, and partially a way to get you to think like a scientist.
Go to conferences or meetings if you can. Meet people. Meet potential projects you can contribute to, and people you could collaborate with. And finally, talk to your supervisor and get ideas.
At the beginning of each week, make a list of everything you have to get done, personal and professional. Make separate lists for long term or short term tasks. When you finish one, take great pride in physically crossing it off the list. If you want, look into bullet journaling. Also, put EVERYTHING on a calendar which syncs with your email and across all your devices. I use Sunrise synced to my personal email, my iphone, my desktop and my work laptop.
I'll add more if I think of anything else. But try not to worry too much; you aren't even there yet! Right now, just get some rest, play some video games, go on a nice holiday. Celebrate your acceptance and look forward to the adventure you're about to have.