The honest headline: a well-made hardwood cane needs very little from you. Wood has been the material for canes for centuries precisely because it holds up. But "very little" isn't "nothing," so here's the short list.
Day to day
Nothing. Carry it, lean on it, hang it by the door. If it gets rained on, wipe it down with a dry cloth when you're home — that's the whole routine. What wood doesn't love is staying wet or living somewhere extreme: don't store it against a radiator, and don't leave it in a hot car trunk for a week. Big temperature and humidity swings are harder on wood than daily use ever will be.
A few times a year
Give it a once-over with a soft cloth and, if the surface is starting to feel dry, a light application of a furniture-grade oil or wax made for finished hardwood. A little goes a long way — you're feeding the surface, not refinishing it.
The tip
The rubber tip at the bottom is the one part of a cane that's genuinely a consumable. It wears like a shoe heel. Check it every few months; when the tread flattens or cracks, replacements are inexpensive at any hardware store and take ten seconds to swap. A worn tip is a slipping hazard, so this is the one maintenance item not to skip.
About the color change
Over months and years, laurel wood deepens — the honey tones shift warmer and richer where your hand lives on the grip. That's patina, and it's the wood recording its time with you, the same way good leather and raw denim do. Don't sand it back, don't seal over it. A cane that looks lived-with looks better than one that looks new. That's the point of buying something built to be kept.