Hey all, chiming in from the Peach forums. My experience with this matchup is rather limited, but I figure I might be able to say a few things...
Barring turnips, you tend to outrange Peach. So I feel like this matchup depends on how well the Lucina player can space Peach out with her normals. Lucina's utilt, ftilt, jab, fsmash, and shorthop nair/fair can punish a Peach who is floating low and trying to use aggressive aerials (as so many like to do).
In air-to-air situations Lucina's aerials outrange Peach's, with the exception of Peach's fair which has massive range but a rather laggy startup. If the Peach player gets greedy and goes for one you can quickly shorthop nair or fair to hit her during its startup. In other air situations (if the Peach is nair-happy, for example), it's just a matter of out-spacing Peach. Peach's nair is extremely fast, doesn't really lag at the end, and has a lot of priority, but none of that matters if she's not in range
A few sidenotes: don't be too scared of Peach landing on you with her dair. Your usmash flat beats it. A lot of Peach players are used to her dair beating out anti-air's during her landing, but this is not the case for Lucina. Usmash her and laugh at her for trying.
Also, don't forget that Peach has toad as a counter move. Though it's not the quickest counter in the world, it still needs to be respected, as Peach can punish obvious dash attacks or tilts/fsmashes with it. Toad can surprise you if you aren't expecting him, and he kills at around 120+, so it's something work keeping in mind.
Speaking of counters, get used to the animation of Peach's fair so you can counter it every time. In general, Peach's fair is an incredible kill move, but it's slow startup means that she usually can't combo into it. It's massivo hitbox and long startup animation (Peach raises her hand to her head, removes her crown from her head, and then swipes at you) means that you can counter it quite confidently.
Oh, also watch out in air-to-ground situations when landing against Peach as her usmash is particularly brutal. If the Peach player positions herself right her usmash will hit you before any of your aerials can, perhaps barring dair (which given my limited experience with this MU seems to trade with usmash, but perhaps it would win if the Lucina spaces really well; but Lucina's dair is risky to land with for others reasons [mondo lag afterwards]).
If you get anti-air sweetspotted by Peach's usmash, you probably die. It kills at dumb early percents, and Lucina is not particularly heavy. I killed my friend (who plays as Lucina) with an anti-air usmash at 80% the other day.
That's all I got! I hope it's helpful to someone