I have enough Peach experience to write a book.
First, know that Peach can own your shield with her air game. ZSS sucks in-shield and Peach owns at owning shields, so your goal should be to not shieldoften. Also, don't roll or spot dodge very often, cause Peach destroys that crap, especially ZSS' ****ty back roll. Just don't do it. Do not pass go, do not collect $200.
In the neutral position, Peach wins. Side-b cannot be used to camp because Peach can powershield and grab, and not only that, but powershield and fair you. In the air, side-b isn't safe either because it can be dodged or avoided and fair'd. Side-b is not a safe camping or spacing tool in this matchup.
Back air is not safe, either, not completely. It's a good spacing tool but if you don't hit, you'll get PS->grabbed. Peach's grab is slow at the back end but the range is pretty good, so I will reiterate: don't spam back air to space. You'll get punished.
Your best tools vs Peach are Neutral Air, Up smash, up air, and down smash. Peach's grab has 22 frames of ending lag, so it's pretty easy to punish a missed grab with down smash. Actually, a lot of Peach's moves have back-loaded lag on them, making dsmash punishes easier than in a lot of other match-ups. As usual though don't spam it.
Neutral air is good because while it doesn't beat Peach's dair clean, it hits at the perfect angle to knock her out of the float. Contrary to popular belief, the top half of peach's body is vulnerable during her dair and nair has the perfect angle to kill it. Snakeee would probably say to just uair it instead.
However, good Peach's won't miss with dair. It's a punishing move and they won't typically just throw it out to try to catch us. Instead they will wait until you are in a weird spot or bad position and then do it, making beating the move impossible. Up air doesn't beat it clean either; it wins, but it's such a tight space that it isn't all that practical without hours of practice. You have to know Peach's dair better than you know your own uair. The hitbubbles on that attack are HUGE, uair's only saving grace is that you don't have much actual hurtbox. Though uair does marginally beat dair, it is a risk you are taking.
Beacuse dair is relatively safe, Peach will want to jump (NOT float, good Peach players do not spam float) quite a bit. While dair isn't beaten by our upsmash, it is so fast that it's easy to catch Peach in it, and it's a great move in this match-up, even better than uair sometimes.
Bair is really nasty, btw. It comes out on frame 6, so it's just barely faster tahn our bair. Uair beats it but again, bair is a punisher and not often used as an approach. You will probably get hit with this attack a lot.
In review, kind of:
1. Peach is laggy on the back end of her moves, so this is a bait and punish match. Punish with grabs and dsmash, but don't screw up.
2. Peach's air game is solid and reliable. Punish predictable aerial activity with upsmash, but for dair, try a uair or nair instead.
3. Do not roll or spot dodge. Instead, try to simply run away or jab.
4. Uptilt does not beat dair. I don't know who started that rumor, but it isn't true. If you want to beat dair, practice your SH uairs and falling uairs and be really ****ing good at them.
5. Turnips are pretty easy to breach. They are slow and easily caught. However, if Peach pulls a turnip you can actually down-b kick her before the animation ends, if you have good reflexes, lol.
6. Peach edgeguards ZSS really well. "But you can down-b the turnips!" Yes, but no. Save your second jump for your recovery and aim for ledges. Do not land on the stage, and don't try to attack her when landing.
7. Peach is easily juggled. This means a lot of free damage when you get her into the air. Getting her into the air is really hard though, if you're playing anyone good. Your opportunities will come from outsmarting Peach, because you have no character advantage that you can exploit.