The basic approach is a CD4017 Johnson Counter and a pushbutton switch. That gets you an electronic circuit that steps through a pattern with each button press. It has 10 outputs, and any of them can be used to reset the pattern. For example, the pattern can be shortened to only 3 stages with a simple wiring connection.
The 4017 has an input that is perfect for a simple pushbutton control signal. After that, it's all about the display. To start, how large? How many directions? Single little LEDs, small clusters of LEDs, etc?
ak