I've dyed (cotton crochet thread) with much success with Rit myself. But there's an even better dye called Procion MX that won't fade as much (if at all) over time. Properly done, your results will turn out as close to professional as can be achieved at home.
I think I've posted this
before, perhaps one day we can stick it on a list of "Must-Have" links or something?
If you opt to use Procion MX, if I can make a suggestion that you also pick up some Synthrapol (a prewash and post dye detergent) AND Retayne or similar product to help set your your dye permanently. You can find suppliers for all of these items through the links within the site referenced above.