Actually, not much care is required. If the soil is fine, and if there is enough water in spring, and the rabbits aren’t too greedy, then all goes well by itself.
Some extra care is required for the young seedlings and the very young plants that have emerged from chipping or twin-scaling. They should not be allowed to freeze (i.e. the earth in the pots in which they grow should not completely freeze). These are also, especially during growing season, more sensitive to dehydration than fully grown bulbs.
Regular inspection of the plants and bulbs is necessary, in particular to prevent mold damaging the plants. For the regular enthousiast it is no longer possible to use any fungicides. So… be quick if something gets moldy. Throw away they infested material as soon as possible! And if you see any slugs, well, then you have a problem as well. Luckily, there are hundreds of ways to tell these creatures that Galanthus does grow for you to enjoy, and not to satisfy their hunger.
If you have received new bulbs or bought them, it is actually a good idea to not plant them in between your other valuables in the garden straightaway. Rather put them in a remote corner for a year. If they turn out to be diseased, at least they can do no harm to your entire collection.
Do not use fertilizer (perhaps a little liqued fertilizer is ok), and it is sufficient to apply some humus in the places where you grow snowdrops in fall. A little lime seems to help. Possibly this stimulates the Mycorrhiza. See also “Feeding and fertilizer“.