Finite state automata (FSAs) sound complicated, but the basic idea is as simple as drawing a map.
This fun activity is based around a fictitious pirate story which leads to the unlikely topic of reasoning about patterns in sequences of characters
An older version of this activity can be downloaded in PDF format here. The content is similar to the current version, but there's some extra technical information.
The Mathmaniacs web site has a similar activity (lesson 5), and a related one (lesson 6)
SwissEduc has a programming environment called Kara (requires Java installed), which is a programmable ladybird robot that has been developed to help teach Computer Science concepts such as finite state machines, turing machines. The variations of this program are given below:
Merchant Taylors' School UK in it's free course on Moodle aimed at AS Computing has an activity in Finite State Machines based on Kara . Read also their Getting Started with Kara guide.