How do I sync my lists across several of my own devices?

Sign in with the same pon account on every device (via your profile — the icon at the top right of the list overview) — then your lists are on the same state everywhere. Without an account, everything stays local on each device.

What pon has learned (order, pantry, locations) isn't synced along with the lists, but can be moved to a new device via iCloud backup.