It depends what you are trying to dye, and whether you need it to be permanent. Food coloring works on most anything if you don’t need it to last too long.

I buy bottles of Rit dye and paint it onto things with a paintbrush. I have also made dye out of watery paint.
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References :

If you are dyeing food or something else to be eaten then food coloring, either liquid or paste, should be used. Mix the coloring with warm water to the shade you want (test a drop on a white paper towel) and then add 2 tablespoons of white vinegar to it.

For decorative items you can thin acryllic paints down to a wash with water to dye things with or use fabric paints. Don’t however use these on anything like plates or cups that will have food on them.
References :

You can use food coloring to dye wool, or nylon, or silk, but you CANNOT use it to dye cotton fabric or acrylic yarn, because it will just wash out. If you dye cotton with food coloring, don’t ever wash it, not even once. You can’t set the dye in any way on cotton or acrylic. It’s like using watercolors.

To dye wool with food coloring, soak it in water with vinegar, then sprinkle on food coloring, egg dye, or artificially colored drink mix such as Kool-aid, then heat your dyebath with the wool in it on the stovetop or in the microwave to help the dye set. See the link for detailed recipes.

You can also use acrylic paint on clothing, but it will feel stiff and scratchy after it dries unless you add "fabric medium" or "textile medium". Real fabric paint gives much better results.
References :
Tie Dye with Kool-aid:
http://www.pburch.net/dyeing/FAQ/drinkmix.shtml

Using Food Coloring as a Textile Dye for Protein Fibers:
http://www.pburch.net/dyeing/fooddyes.shtml

Fabric Paints: a different way to color fibers:
http://www.pburch.net/dyeing/fabricpaints.shtml