The manufacturer of most every tool I have for screen printing and block printing is called Speedball. They make a starter kit which I think might be about $50 or $60 with a screen and squeegee and a few types of acrylic ink for paper and fabric. It's a good value and makes it all very convenient. Ray got me the kit for our first anniversary and I was sooooo excited, but it still took me over a year to do anything with it because I was really intimidated and thought it was complicated. It's not! I'll simplify it as much as I can.
The basic concept is to pull ink through a screen which allows ink to flow through, except in areas which are otherwise blocked off. So if you cut a 4x4 inch square out of a piece of paper and laid it down under a screen and pulled ink through the screen, there would be ink everywhere except where the square is. So you need a stencil. So get yourself a 8.5x11 piece of paper and a cutting mat and an exacto knife and cut something out. Let's say a sunburst, since that's what I did today. I used a transparency so I can re-use it (the ink will soak your paper and so it's not re-useable.) Here is my sunburst stencil:
You can trace an image or create your own. I drew something I liked then laid my transparency on top and traced it with a sharpie, then cut it out. Don't forget to think about positive and negative space! The holes you cut out are where you'll get ink on your shirt.
To set up your printing, you need a pretty good work space. I'm lucky enough to have this old tool bench in our basement as my permanent set-up.
Paper towels, tape, ink, plastic cups and spoons for mixing, a screen and squeegee, a bag for trash, and you're all set. It's also good to do your printing on top of a soft surface so you can apply pressure as you're pulling your squeegee down your screen and the screen can give a little. I have 2 layers of felt on the countertop, covered with newsprint. Give yourself at least an hour to do your first print and get a feel for set-up, printing and clean up.
You can use color right out of the pot, but usually I add some white and a tiny bit of black for more depth. I also added a little yellow to my blue to mellow it. So tape up the sides of the screen so that no paint can get into the edges of the wood and so that there is no open space for ink to flow through, other than where you've cut out.
Here's the screen ready to go. The stencil is underneath the screen next to my fabric. Scoop some ink across the top of the screen and hold your squeegee at an angle toward you and pull the ink down toward you across the screen. You'll want to do one or two test runs on a scrap piece. I have a piece of muslin I use to test everything. It's fun to look at things I've done in the past! 
Yes, that's a Streetcar and a red bean! It takes a few runs across the screen to bring the ink all the way through the stencil. But it only takes one time dragging the squeegee across the screen for the stencil to stick to the screen. So you won't need to reposition your stencil every time you lift the screen. Since I rarely do many multiples I don't worry too much about "flooding the screen" but if you're doing a bunch of prints, you should learn more about this. I think it's basically putting some ink on your screen to keep it wet between prints. Once I'm done I always hurry to wash the screen off so I can reuse it. I've don't at least a dozen prints using this screen and hope it'll last me awhile longer. Here's the outcome of today's project. I'm thinking about adding another color and image on top of it, but don't know what that'll be yet.
I hope this is helpful! Screen printing is definitely not the simplest printing to do, but it does give nice crisp images and you can do a nice run of a lot of images in one session. But it's difficult to use a large piece of fabric. I hope you have enjoyed this and it's clear enough for you to follow. I'd like so answer any questions or see pictures of projects you try.
Also, I recommend anyone who is interested and doesn't know where to begin to buy Lena Corwin's book Printing By Hand. It is completely amazing. I will blog next about block printing, which is a super-easy fun project to do.
