I somewhat understand finite state machines in concept however am unsure how I would apply it in this example. The question is
Draw the transition diagram of a finite state recogniser which accepts precisely those strings which satisfy the regular expression :
$c(((ab) \cup(bc))a)^*$
These kind of problems are puzzles of sorts. The first thing I would do is to consider what strings are contained in your regular expression. Since the star operator allows repetition of zero or more of the symbols in its argument, we have that $c$ is an acceptable string.
Also, the parantheses inside the star operator $((ab)\cup(bc))a$ gives you the substrings $aba$ and $bca$, and since any string has to start with a $c$, both $caba$ and $cbca$ are acceptable strings.
Because of the star operator, in fact any repetition of $aba$ or $bca$ after the $c$ will be an acceptable string, e.g.
$$cabaaba$$ $$cbcaaba$$ $$cababcabcaaba$$ etc.
So I need a machine that ends up in an accepting state when it is given a $c$ as input, and from here, any succession of the strings $aba$ and $bca$ should end back up in the accepting state. Any other input should be trashed.
In my drawing below, the middle state (marked "S") is the starting state (which I signify by the arrow leading into it from nowhere), and the bottom middle state (marked "A") is the accepting state. I use the top state for trash, i.e. whenever I come across a symbol I don't like, I redirect the machine to the trash state. Once you are in the trash state, you can never go back. The trash state cannot be escaped.