How do we figure out distance travelled.
18.0m(W) in a time of 6.00s and stopped. After stopping, it moved 12.0m(9E) in 10.0s

How do we figure out distance traveled?

moved 18.0m(W) in a time of 6.00s and stop
After stopping, moved 12.0m(9E) in 10.0s

Distance = Rate x Time.

You are given two distances and two times. You can solve the above equation for rate and get that:

Rate = \frac {Distance}{Time}

\frac {18\,m}{6.00\,s}

\frac {12\,m}{10.0\,s}

I'm afraid I don't understand your question, "How do we figure out distance traveled?" The distances are given, 18.0 m and 12.0 m. The sum is 30.0 m and that's the distance you traveled. If you are after an average velocity (average rate), take the total distance (18 m + 12 m = 30.0 m) and divide by the total time (6.0 s + 10.0 s = 16.0 s). The average velocity is, therefore

Average\, velocity = \frac {30.0\,m}{16.0\,s} = 1.875\,\frac {m}{s}

Do you mean that the velocities are:
18 m/s for 6 s towards the west, and
12 m/s for 10 s towards the east?

Then, the distance travelled is 18+12 = 30 m
The displacement is 18-12 = 6 m to the west (taking the western direction as positive)
The average velocity will then be 6/16 = 0.375 m/s