Defining if a language is regular
Hint: Here are a few easier languages to practice on:


Every occurrence of the symbol $a$ is followed by a non-$a$ symbol. (Hint: every $a$ occurs in the context of $ab$ or $ac$.)
No occurrence of the symbol $a$ is followed by another $a$. (The only difference from 1 is that a word could end with $a$.)
Every occurrence of the symbol $a$ is followed by a non-$b$ symbol. (Hint: what are the possible contexts of $a$?)
Your language. (Combine the answers to 3 and 2.)