If you want Inkscape to remember your last used style, go to Edit > Preferences > Tools > Pen, and select the option that says "Last used style". Now when you change the stroke width, Inkscape will remember what you set it to, even if you quit the application. You can also apply this option to the other drawing tools, such as the Pencil, ...
You could use Inkscape which is free, or virtually any vector image editor.
Copy and paste your signature image into Inkscape, lock it in place. Then draw the signature paths on top using the Bézier Tool (aka Pen Tool). Once you have finished unlock and delete the image.