Once a year: the full service
This is what most people do in spring, and that makes sense: that is when the appliance has stood idle longest. It concerns the parts that are not ceramic, because those are the only ones that genuinely wear.
Gaskets: squeeze them back into shape before replacing. A fibreglass gasket gets flattened through use, and the first thing people think is that it needs replacing. Usually it does not.
Our gasket has a brass sleeve inside it. If the gasket has gone flat, squeeze it so that sleeve comes back into shape. The gasket regains its volume, the lid seals as it should again and closes noticeably more softly. That is a minute of work and it saves you a replacement.
Replacing is only called for when the gasket is genuinely worn out: frayed, coming loose in places, or no longer able to be squeezed back into shape. Why this matters: a gasket that does not seal means air comes in along a route you are not steering, and then you can no longer control the temperature precisely. You notice it mostly in low temperatures you can no longer keep low. See also How do you control the temperature in a kamado?.
Hinges and bolts. A year of temperature swings shakes bolts loose. Check whether the lid still hangs in balance and adjust where needed. Moving parts get silicone spray, not scented oil: that soaks into the ceramic and into your food.
Cast iron. The top controller is the part where damp takes hold fastest. Work rust away with a BBQ brush, not steel wool: that is too aggressive and leaves scratches in which new rust settles. Finish with linseed oil.
Wood and exterior. Wipe the side tables and the handle with lukewarm water, then dry thoroughly. A treatment with outdoor furniture oil belongs here. The ceramic exterior has no pits where dirt can lodge, so a cloth does the job.