Moss is not a fast plant, and that is worth knowing before you start, so you neither give up too early nor expect a carpet by next month. Here is what to actually expect.
Realistic timescales
Kept constantly damp and shaded, a slurry or fresh transplant usually starts to green up and grip within a few weeks. A patch knits into continuous cover over a few months. A convincing moss lawn or wall takes a full season or two to look established, and the deep, springy cushions of an old moss garden are the work of years. None of it is quick, but most of it is steady.
What speeds it up
Three things, mostly: constant moisture, deep shade, and firm contact with the surface. Mild temperatures help, which is why moss does much of its visible growing in the cool, damp shoulders of the year, spring and autumn, rather than high summer. Soft rainwater, a humid spot and protection from drying wind all push it along.
Why it seems to stop
Moss switches off when it dries out, shrivelling and going dormant until the next wetting, so a patch that looks dead or static in a dry spell is usually just waiting. It has not failed; it is paused. Resume the moisture and it resumes growing. The single biggest mistake is deciding it has died and giving up during a dry fortnight. Patience is most of the technique; see the growing guide.