Run water over the finished piece to remove any remaining dust. Allow it to dry thoroughly before you begin painting.
Many different types of paint can be used to fill your pins engraved areas. My personal preference is Golden High Flow Acrylic as it doesn't need to be thinned with water before application. Alternatively, gouache paint offers some advantages with saturation and opacity – but it can create issues with viscosity and air bubbles.
For the level of detail in my bird design, a 2/0 "round" brush will work fine. For pieces with finer detail, I'll often use a 20/0 "spotter petite" brush.
Fill the engraved areas of your design thoroughly with paint, making sure to cleanly fills each shape's corners and edges. The High Flow acrylic paint will flatten a lot as it dries - fill each engraved shape up like a small pool/puddle to avoid the board's FR-1 material from showing through.
Once painting is complete, allow your piece to dry thoroughly in a dust-free area. I place mine in a resealable food container with the lid left ajar to provide some air circulation.