There's many ways to decorate a t-shirt. The easiest is by printing your computer-designed graphic onto an iron-on transfer and ironing it onto your shirt of choice. They've come a long way from the mirror-image transfer sheets my mom used to put on my T-ball uniform. Now you print in "positive", the colors are vibrant, picture sharp, and will work on dark shirts (or other mostly cotton textiles). Look for "Dark T-Shirt Transfers" (Avery is one brand). Office Max or Staples should have them where a kit with 5, 8x10 sheets costs about $10. This is how I've done shirts in the past.
As great as those are, you are limited to very simple shapes (although what you print on it is limitless), they don't do too well after being washed several times, and they just don't have the same 'feel' as a professional silk screened graphic. The method that follows is much closer to silk-screening and, because you are using the same inks, are just as durable. This is also the cheapest method, assuming you already have a few things.
What you'll need
-Silkscreen fabric paint - found at art and craft stores. Speedball is the leading brand. ($6)
-Paint brush - it should be as wide as the largest area to be painted ($2)
-parchment paper - used for baking, found at good grocery stores. ($3)
-Freezer paper - also called butcher paper. My market didn't sell it so I begged a butcher for some. (free)
****note: Wax paper won't really work. Freezer paper has one side non-waxy so you can print on it.
-Graphic / Design
-Household Steam Iron and ironing board
-ruler or straight edge
-blow dryer (optional)
-A printer and some way of printing the graphic (i.e. web browser, photoshop, illustrator, word)
*note: You could alternately hand draw the logo