Attaching table legs to a round table top isn't difficult; the challenge is making the leg attachments sturdy enough to last a lifetime. The trick is to prevent side-to-side movement that occurs when the table is slid. By building an apron for the table, you can provide the proper bracing so your table will remain sturdy.
Measure and cut four lengths of lumber forming a square that fits inside the radius of your table. Using the 3/16-inch bit to pre-drill two holes 3/4 inch apart, 3/8 inch from the end, on both ends of the two longest boards. Glue and screw the apron frame together.
Drill eight holes equally spaced, two per side, down through the edge of the apron using the 3/16-inch bit. Countersink with the 1/2-inch bit so that the sharp end of the screw does not break the surface of your table when you screw on the apron. Check this by placing the screw in the hole and measuring how much the sharp end protrudes out the bottom of the apron.
Turn the frame upside down and run a bead of glue along the edge. Flip it over onto the bottom of the table. Measure from each corner of the apron to the edge of the table to position it, then screw it down.
Place your table legs on the inside corners of the frame so that two sides of each leg have contact with the apron. Mark the locations for screw placement. Remove the legs and drill with the 3/16-inch bit. Glue the surface where the legs contact the frame. Put the legs in place and screw them to the frame.
Use a framing square when attaching legs.
Always wear safety glasses when doing carpentry work.