One simple way to make vector paths in Photoshop is to convert them from selections:
- Use the selection tools (mainly Magic Wand, Quick Selection, or Lasso) or the Select > Color Range command to establish a selection.
- Use the Paths palette > "Make Work Path From Selection" command (next to "Create New Path").
- Select the path using the Direct/Path Selection tool (A).
- Add a Solid Fill Layer (Layer > New Fill Layer > Solid Color), and Photoshop will automatically use the path as the layer's vector mask.
Paths in Photoshop are normally invisible, and need to be filled, stroked, or become part of a vector mask to have a visible effect.
Photoshop isn't quite a vector program. While it does have vector tools, it ultimately produces bitmap formats (e.g. JPEG, PNG).
You should consider getting Adobe Illustrator, which is designed for vectors and can work in CMYK mode for printing, or Inkscape, which is a free vector editor. T-shirt factories should be able to handle the AI and SVG formats, but you should ask them to be certain.
Tracing tools like vectormagic.com, or those available in the aforementioned programs, are not perfect, but can carry you pretty far. You'll have to learn how to manipulate paths in the chosen program to get rid of "blur" (typically turns out to be very thin lines of color, a consequence of the automatic vector-tracing process).