Do you really think Andy did his own hat? HAH!
I agree with using a straw hat, it's much cooler. For painting the hat I used Krylon exterior spray paints, and for the detailing I used acrylic craft paint, then I used a coat of spray polyurethane over the whole thing to protect it.
If you want to do stenciling of the logo or anything, contact paper works really well, as it sticks to the hat just enough for you to trace around, but is easy to remove, and won't take off any paint that's underneath it. Sticky backed craft felt also works well for more permanent logos, and the colors won't run. I used it on my first hat, and it worked really well.
If you need fabric or ribbon, Jo-Anns or Hancock Fabrics usually has some really cute red, white, and blue patterns around the fourth of July. (If you want to decorate with beads, PM me. I've got a TON of pony beads left over from my last hat that you can have if you want them).
If you want to go sparkly, you can also find the plastic flat backed rhinestones at craft stores. (Just a note: I found that blue is really difficult to get a hold of).