I'd use a spirit-based stain, or maybe oil-based, rather than a water-based stain for MDF. Unlike paints, stains are intended to penetrate relatively deeply, and when water penetrates MDF it swells and distorts.
Spirit stains dry a lot faster than oil stains, but generally require a protective top-coat of oil or varnish. That's less necessary with an oil stain, but in general it's still a good idea.
You can get a variant type of stain which is usually marketed as "varnish stain", which is a transparent pigment suspended in a polyurethane or acrylic medium. They act more like paints than true stains; they can be really handy for "dip" glazing effects.
The specific brand names will vary depending on where you live.