What size should you make your child's furnishings?
The key is the size of the chair. The seat of the chair should be just high enough that the child's feet will sit flat on the floor and the child's knees will be bent at a 90-degree angle. At this height, the chair will be comfortable and promote good posture.
Once you know the proper chair size, you can design the table. The top of the table should be about 8 inches above the seat. Combined with a properly-sized chair, this table will give your child a comfortable place to study and play for many hours.
The simplest way to build a chair is to use plywood, rather than making many small pieces to be assembled. Plywood sheets are large enough that you can make a chair from one sheet, using as few as four pieces. These are the back, the seat, and two side pieces which combine the legs and the armrests (if you need armrests). The back serves as both backrest and brace, making a sturdy chair that will not wobble. This piece can be a square or rectangle; the same is also true of the seat.
The sides are your opportunity for creativity. You can shape individual legs or make one large base; you can make the sides tall enough to serve as armrests, or only tall enough to support the seat; you can make a tall back or a short back. The only real rule here is that you should make the back edge of the sides angle back behind the child, to provide a wider base so the chair will not tilt over backwards.
Tables can also be made entirely from plywood. Construct the table like a box with one side open, but with leg-holes cut in the four sides. These four sides will then resemble inverted 'U' shapes.
You can also use a plywood top with regular lumber for legs. You can build an apron of 1x2s or 2x2s to strengthen the table top and aid in attaching the legs; or you can buy premade legs that attach with specialized hardware.
Nails and glue can be used, but screws and glue will make stronger joints that can better stand up to rough treatment. Use a drill to make pilot holes for the screws to minimize splitting.
Tables and chairs can be finished with almost any finish: stain, clear-coats (like shellac and varnish), or paint. Paint is probably the most popular, not only because it is colorful, but because latex paints are easy to clean up with soap and warm water.