There's no quick and easy way to explain how to use vector tools. Some advice:
You can use the Pen tool in Photoshop to create either a path or a shape layer of just the lines.
Do a search for "Photoshop pen tool tutorial" (w/o the quotes) on google, and you will get a ton of good hits. BTW, a really good one can be found at
http://www.gurusnetwork.com/tutorial/pen/ . Work through those and practice, practice, practice. The pen tool is NOT EASY, and getting good at it will not happen overnight.
There are actions that will turn the raster image that you have into a vector path, but none of them will get the quality that doing it manually will.
Sorry, that's about all the time I have to really help you out here.