Hey bud: instead of painstakingly tracing the image into Vector... for something like this (a stencil), go use a free round at VectorMagic.com.
Seriously... that must've taken a long time if you aren't a professional (it'd still be a while if you are), and it could've been done in about 3 minutes for free. Just in case you want to play with that kinda thing later.
Oh; it only gives you a few credits per IP and Email. So if you sign up with an e-mail address and then sign up from the same computer/connection with another e-mail address, it won't give you more free credits. You need a new e-mail address and a new IP. If you don't know how to do that, just ask.
In computer graphics, vectorization is the conversion of raster graphics into vector graphics, a process also known as image tracing. What vectorization does, is it takes and image that has a defined size and pixels, and it groups sections together, much like polarization. This allows us to scale an image infinitely.
That being said, I usually use a website called http://www.vectormagic.com/ to vectorize the images, then download the .svg, throw it into illustrator where I will then export it at a much larger size .png and then i am able to use it in photoshop.
Please note that these images will NOT look the exact same, as colors are grouped.
Edit Also, on that website they only allow two free conversions. To get around this, I would suggest using a temporary email account to sign up within a proxy because they track the IP address to confirm if you've already made an account.
Turn the stencils into vectors first here that way you can blow the images up to any size you want without loss of quality. Then you can use posterazor or rasterbator to scale it up and divide it into manageable chunks.
Why not just scan the artwork and autotrace each piece in Illustrator to vectorize? If it's simple line art, you should have no problem using the pen tool even in photoshop to create outlines. There is even a tool called Vector Magic that will automatically convert line art (or anything else) into vector format, and exports to .eps, .svg, and .pdf formats. It's pretty accurate with line art, comparable to Illustrator's autotrace.
There is no button in any Photoshop program that "makes this higher quality"... That is impossible. What you might want is a vectored image which may look a little cartoonish, it's what most logos are, but if you are interested in seeing what you might get from that process, try http://www.vectormagic.com/home and upload your image