Summarize:

Since you will be painting on your plywood, you want it to have a smooth texture. Signboard is a great option, but if you can't get it, sanded plywood or medium-density fiberboard will also work. A common size is 4 by 4 feet (120 by 120cm), but you can make your barn quilt whatever size you want. A smaller one is an easy way to start. A lumberyard or a home supply store are good places to look for plywood. Do your priming and painting outside or in an open garage. Lay down drop clothes to protect the floor if you are painting on the ground. You can also put your plywood on top of sawhorses or on a work table. If you don't have ventilation when you paint, the paint fumes can be really harmful. Pour primer into a rolling tray, and dip your paint roller into it. Rub the paint roller along the plywood in smooth, even strokes. It's important to use an exterior primer, since barn quilts are normally hung outdoors. Allow the coat to dry, following the instructions on the primer. Different kinds of primer will take different amounts of time to dry, so check the instructions on your can of primer to figure out how long you need to let it sit. Then paint on the second coat just as you did the first one. Let the primer dry again. You need to prime the back of your board as well, even though you won't be painting it. This helps keep your artwork waterproof, since the barn quilt will have to withstand the weather outdoors. Be patient and wait for the primer to dry in between each coat. It will help your quilt last much longer.
Buy a square of smooth, pressure-treated plywood. Work in a well-ventilated spot and protect the floor with drop cloths. Apply exterior latex primer to the plywood with a roller. Let the primer dry and apply a second coat. Paint 3 coats of primer on the back (rough) side.