Painting clay means three different jobs depending on what state the clay is in, and the paints do not swap. Colour on unfired greenware is usually slip or underglaze; colour on bisque before glazing is underglaze; colour on a dried air dry body is acrylic. Choosing from the wrong list is the most common reason a finish fails.
On greenware, colour goes on as slip or underglaze
Unfired clay that will be fired later takes coloured slip while it is leather hard, or underglaze once it is dry. Both are ceramic materials that survive the kiln, and both bond to the body rather than sitting on it. Acrylic on greenware burns away in the firing and can leave a mess in the kiln, so paint for clay that will be fired is not paint in the household sense at all.
On bisque, underglaze then a clear
After the first firing the body is porous and takes underglaze cleanly, which is why most decorated ware is painted at this stage. The colour is then covered with a clear glaze and fired again, which is what makes it permanent and washable. This is the route to a finish that survives use, and it needs a kiln at both ends.
On a dried air dry body, acrylic and nothing else
There is no firing, so ceramic colour has no reason to be there. Acrylic over a primed, fully dry surface is the whole method, sealed with a varnish if the piece will be handled. The finish is decorative rather than functional, and no combination of paint and sealer turns an unfired body into something that holds water.
Where paint is the wrong question
If the object is meant to be used, the finish is a glaze rather than a paint, and the decision is made much earlier: buy a fireable body and arrange a firing. The suppliers priced on this site sell those bodies from roughly a pound a kilogram excluding VAT, and studios that fire other people's work are common. Painting is what you do when firing is not part of the plan.
Questions people ask about painting clay
What paint for clay can go in the kiln?
Underglaze and coloured slip. Household paints burn away and should not go into a firing.
Can I paint pottery with acrylic instead of glazing it?
You can, and the result is decorative rather than functional. It will not be washable or food safe.
Do I paint before or after firing?
Ceramic colour goes on before, usually on bisque, and is covered with a clear glaze. Acrylic only ever goes on an unfired body that will stay unfired.