As a former screen printing company owner and designer, I can tell you the industry standard is CorelDraw. The color separation and basic CAD technologies that were adapted early on are still universal today. Walk in to any screen print, embroidery, ad specialties, sign or vinyl banner shop with a CorelDraw .cdr file and you're good to go.
The downside is that CorelDraw is the least desired vector app by professional designers because it is imprecise, the colors are dull and not quite right, and lots of other flaws. For screen printing, any vector drawing program would probably do, even Inkscape, which is free. If you use heat transfers, any design app will do. Paintshop Pro is a good one for hobbyists. Photoshop Elements is better but a little more expensive, but Paintshop Pro is owned by Corel, so is CorelDraw friendly.
Be advised that screen printing is expensive, labor intensive and has a high spoilage rate. Designing, preparation and clean up for screen printing a dozen t-shirts takes hours. The actual printing takes only a few minutes.
As a beginner, because of ink smudges, leaks and other printing mistakes, you might have to print a couple dozen shirts just to get 7 or 8 acceptable ones. If you are not planning on doing this as a profession, or eventually printing dozens or hundreds of something, heat transfer is a better alternative. Not iron-on, although you could start with that to see if you like doing it, but heat transfers applied with a professional heat press machine as is done by cafepress.com or zazzle.com or any local Kinkos or quick print shop.
Oh...and the stencil process? It is not created or applied with software. The stencil material either comes in sheets that you cut and apply to the screen, or is painted on. When the stencil material is dry, you will have to expose your art transparency to some kind of intense light source. Then the screen is washed out with warm water. Any areas blocked by your art will wash out and create the stencil that you will squeegee ink over that makes the final print.
Best of luck.
"Silk Screening or Serigraphy
In 1907, Samuel Simon of Manchester England was awarded a patent for the process of using silk fabric as a printing screen. Using materials other than silk for screen printing has a long history that begins with the ancient art of stenciling used by the Egyptians and Greeks as early as 2500 B.C. A few years after Simon's patent, John Pilsworth of San Francisco developed a multicolor process of silk screening called screen printing. The term "Serigraphy", comes from the Latin word "Seri" (silk) and the Greek word "graphein" (to write or draw)."