Start by drilling pilot holes in the apron strips for the screws. Pilot holes will allow the screw to go through the piece without splitting the wood.

Next, secure the apron strips to two adjoining faces on each leg with wood glue and screws.

Sand each of the legs starting with 80 grit sandpaper wrapped around a sanding block or scrap piece of wood, then sand with 120 grit and finish with 220 grit.
Repeat for the apron pieces before attaching to the legs.