I encourage my readers to solve this for themselves. However, if you give up, you may click the solution button below.
Solution
Here is one possible solution: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 17, 18, 12, 11, 18, 17, 9, 8, 7, 6, 5, 4, 3, 2, 1, 15, 14, 13, 19.
The key is to notice that you can make a u-turn in the 11-12-18 triangle. The solution is to go the whole way around the outside circle, in a counter-clockwise rotation until you get back to 1. Then take the inner circle clockwise to 18. Then make a u-turn and get back to 18. Then retrace your steps back to 1. Then it is an easy path to 19 from there.