The whole care routine for a handcrafted wooden mug fits in one paragraph, and it’s the same paragraph I give every new customer: hand wash with mild soap and warm water, dry it right away with a towel rather than letting it air dry, and never put it in the dishwasher or let it soak in the sink. That’s it for day-to-day use.
Most of my pieces have resin finishes that do not need to be regularly oiled or waxed. Those that are oil-finished just need a bit of oil rubbed in every few months.
Follow that rhythm and a well-made wooden vessel will genuinely outlast most things in your kitchen cabinet.