Inkscape
A free software for editing vector files. One of the free alternatives to Adobe Illustrator.
How to turn an image to a symbol
Import the image into the canvas. We will use the image to trace the lines on it. As a personal preference, reduce the opacity of the image to make it easier to see your work in progress.
Then it's all free from here. For a start, you can use the calligraphy and the bezier curve tool.
Repeating patterns on a shape or path
you need two things: the object (e.g., a shape, a single line) and a pattern to stamp all over the object
select the object and apply the "Pattern along path" effect on it
configure the effect to repeat the pattern
copy the pattern
paste and link the pattern into the object
to finalize the effect, convert the object into path
the new object is a group composed of the pattern tied together, you can break it apart to change each individual pattern