What kind of paint is very shiny and works on plastic?

Epoxy enamel paint.

You will need to prepare the surface of the plastic to hold the paint, or it will easily come off. Lightly roughing the surface with fine sandpaper, or VERY lightly wiping it with acetone will remove the smooth surface enough to hold the paint.

You might try oil based enamels. They can be purchased at an art, or craft store or wal-mart in the paint section. They are pernament, takes about 24 hours to dry, clean-up is with mineral spirits like turpentine or car gas and it comes in pint, quart and gallon cans in a very large quanity of colors.

Acrylic artist's paint

Acrlyic Enamel paint spray

Krylon and Rust-O-Lium both make paints specifically for plastic.