What is the relation between spore, cyst and gemmule? Do they form a continuous chain i.e. spore develops into cyst and cyst develops into gemmule due to unfavourable conditions?

Cysts refers to a collection of cells, like free living organisms that clump together and form a supercolony, or endocyst. These are often seen in certain marine archeae and cyanobacteria, or corals. This colony is not a germinating spore, or in any way associated with reproduction, other than its obvious survival advantages to the organism(s) as a whole.
Spores on the other hand are typically self replicating reproductive bodies or non vegitative states of organisms. The best studied spores are those found in certain types of fungus, bakers yeast (S. cerevisiae) and Neurospora crassis and bacteria of the bacillus sp. which includes the extremely well studied strain B. subtilis, and the pathogen B. anthracies, which causes Anthrax.
Gemmulesare internalbudsfound inspongesand are involved inasexual reproduction. It is an asexually reproduced mass of cells, that is capable of developing into a new organism i.e., an adult sponge.