The brush supplied is usually a small one, to enable you to paint the smallest shapes in the painting. It can make painting larger shapes very tedious so, if you've got a bigger brush use this too.
Start with either the darkest color and end with the lightest or the other way around, leaving any segments that have a mixed color (double number) till last. The reason I recommend doing the colors in sequence from dark to light (or the other way around) is that this helps you learn a little about the tone and chroma of colors.The contrast between the white (light tone) of the paper and the darkest color will be very stark. As you add each subsequent color, you'll see how they impact on one another, influencing the way each looks.
Keep a jar of clean water for washing your brush (assuming it's an acrylic Paint by Numbers kit) to hand, as well as a cloth for wiping and drying the brush. Don't dunk the brush into the paint all the way up to the ferrule, just the tip. Rather pick up paint more frequently than have a glob of it fall off onto the painting.
Be patient! Don't splay out the hairs of the brush in an attempt to paint in an area more quickly. This will quickly ruin the brush and destroy the fine tip. Apply gentle pressure to bend the tips of the hairs slightly and glide the brush along the surface. Think of it as the paper (or canvas) pulling the paint off the brush rather than using the brush to push the paint down.