You can use artist's oil paint but it will take a long time to dry.
You can also use "oil-based" house paints, "enamels," etc--those won't take as long.
Generally though, quick-drying and easily-available acrylic paints are used on air-dry clays (sometimes more than one coat is needed, or one coat over a primer) or latex house paint, but really any permanent paint can be used.
If the paint is not permanent (tempera, watercolor), it won't seal the air-dry clay --and sealing is necessary on air-dry clays to protect against later water/humidity-- so you'd have to add a coat of clear permanent finish on top of that paint (or even mix it into a clear finish before painting on).
There's more detail my answer here too:
https://answersrip.com/question/index?qid=20120322132132AAoME25
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