From what I understand (which you may already know...)
The actual process of applying silk screen ink & dyes in printing process only takes a few hours to learn....
The process of making the screens for the printing is what takes a L-O-N-G time to learn.
there are probably many places that will ink t-shirts for you, if you provide the computer files for the 4 color patterns.
I am thinking you would need to start with a class of software call "Vector Graphic" for example Corel Draw Suite, Adobe Illustrator, or if you are brave, Inkscape (Free software with no support and less instruction and help.) Your local community college might have a computer graphic illustrator classes.
Here is an example of on-line tutorial for CorelDraw (I post this because I am familiar with the product, but I am not endorsing CorelDraw to a raw beginner, you will want to buy the product recommended by your night school instructor.)
Corel DRAW tutorial
The add on is called CorelDraw Fashion Factory.
The next step is just printing to a special printer, which is just a machine you buy.
You can use your Adobe or Corel software to also create a command file for a multi-thread sewing machine with up to 16 needles and thread. I offer the following only as an example of what I am talking about. 4 work stations, each with 12 different colors for embroidery of a logo on to a shirt or hat. You can Google on your own for: embroidery multi-head sewing machines.
Brother International - Industrial Embroidery and Sewing
Again, all starts with the computer vector graphic software.
Aside: same with a wood laser cutting "scroll saw" pattern making, stemware (wine glass) engraving, and of course a few CNC brass and metal engraving tools.
For the most part, running the printing machines, is at the bottom of the food chain for earning money. The earning potential is in the artistic / graphic design end. Then sub-contract to lowest bidder for the grunt work of actually squeegeeing the paint or dye across the t-shirts.
This post could go on for a long time, so I will hold off for now.
Phil