I have only ever used acrylic paints on all my projects. I have just recently started using glycerin with acrylic paint. It is used sort of like a liquid clear. This has been helping keep the paint wet longer especially on wood.
I have painted on cheap wooden boxes and plaques from a thrift store and from a craft store and I really have not noticed a difference in the end product. With the exception that the hardware usually is a better quality at a craft store.
I have painted wet on wet technique on these wood surfaces many times. I do not prepare the surface before I start but I do make sure it is smooth. I take off all hardware before starting. I simply follow along with Bob. **be sure to check before starting that your lid is positioned correctly** I unfortunately made the mistake once of painting upside down. I didn't notice until trying to put the Hardware back on. Very disappointing. I had to make new holes for hardware and you can see the old holes.
I have always coated the boxes afterwards with varnish or Modge Podge ( which is a blend of varnish and glue and water) This seals and smooths the surface. If there are wood slivers while painting, it goes away after varnishing.
These are so fun to do. I hope that you try it too and show your work.
If you have any questions feel free to ask and I will try to answer as best as I can.