You can use any type of paint, it is the primer that is really important. Fortunately, there are several on the market that will work.
Zinsser makes very good primers. Both Cover Stain (oil base) and 123 (acrylic) will work well. These primers are white, but they can be tinted to a pastel color. Another very good primer is XIM. They have some of the best primers for plastics on the market. Plus they come as a clear or "Can be tinted to deep colors".
Both acrylic and oil based paints can be applied over these primers. The best paint, in my opinion, is an oil based enamel with a semi-gloss sheen. The biggest drawback is a long cure time, 2-3 weeks in most cases. Plus it is a little difficult to work with. A good alternative is a 100% acrylic interior semi-gloss enamel.
You have several options for applying the paint. Plus, the paint could be mixed with a glaze to make it more or less transparent. This is the same glaze used for many faux painting techniques.
The paint, or glaze mix, can be applied with a brush, fine or course sponge (natural or synthetic) or inexpensive sponge paint applicators. Each will produce a different effect. The paint could be applied with a brush then the excess removed with a sponge.
This will take some experimentation to get the look you are after.