Moss does not follow the garden's usual calendar. It does most of its growing when the showy plants are dormant and rests when they are at their peak, so caring for it well is mostly a matter of working with that rhythm rather than against it. Here is the year, season by season.
Spring
Spring is prime moss season. Cool, damp air and steadily lengthening days are exactly what it likes, and growth is at its fastest, so this is the best time to start new moss, whether by transplanting patches or laying down a slurry. Established moss greens up and thickens with little help. The one spring job worth noting is that birds gather moss for nests now, so expect a little pilfering from lawns and walls, and pin down freshly laid moss if they are lifting it before it has gripped.
Summer
Summer is the hard season, because heat and dry air are moss's main enemies. Established moss in deep shade usually rides it out, drying and going dormant in spells without rain and greening up again when the weather breaks, so brown summer moss is almost always just resting rather than dead. New or recently laid moss is far more vulnerable and may need watering through dry weeks, ideally with rainwater in the early morning or evening rather than the midday heat. Resist the urge to dig up a brown patch and give up on it; wait for rain.
Autumn
Autumn brings back the cool, damp conditions of spring, and with them a second flush of growth, which makes it the other good window for establishing new moss. It also brings the single most important maintenance job of the moss year: clearing fallen leaves. A thick blanket of wet leaves left over a moss carpet smothers and kills it over winter, so sweep or gently blow leaves off through the season. Do that one thing and the moss comes through fine.
Winter
Winter is moss's quiet advantage. While the rest of the garden is bare and brown, moss stays green and even keeps growing slowly in mild, damp spells, which is much of why a shaded moss surface looks good when nothing else does. It tolerates frost without harm, simply pausing and resuming. There is little to do beyond keeping heavy leaf litter and debris off it, and enjoying the fact that the greenest thing in the garden in January asked nothing of you.
The rhythm of it
The pattern underneath all this is simple: moss does its real work in the cool, damp shoulders of the year, spring and autumn, rests through summer drought, and quietly carries the green through winter. Plan your planting for the shoulders, your watering for summer, and your leaf-clearing for autumn, and the rest looks after itself. See the growing guide for the establishing basics that all of this builds on.