When the flowers have been cut and the growth in the stems has stopped, the earlies can be dug up, planted in boxes, labeled and stored for the winter in a frost-free area, before being brought into the warmth again in January to start producing the next season's cuttings. The lates can be treated similarly or just left in their pots.
One important point is that no two cuttings produce identical plants, there are many tiny variations and this means that stock selection is very important. If you have three or four plants of one variety it is well worth marking the best two while the flowers are still on, and only keeping those, to provide all the cuttings required the next year.
One problem that can crop up is the difference in colour and type of blooms that are flowered on late buds. In most cases this means that later flowers have a deeper colour. However, some whites may become light pink and some incurving varieties tend to reflex and even become daisy-eyed. This will not affect any cuttings that may be taken for next season's crop, when they will produce the correct bloom if flowered at the right time.