When you're tiling a floor with a drain in it (whether it's a shower floor, utility room or other setting) you should first lay out the tile grid as if there isn't a drain there. That way, you can establish where the tiles are supposed to go in relation to the grid before you cut them around drain. If there's a noticeable slope to the floor leading into the drain, make sure to use smaller floor tile (four inches across or less), so you can more easily follow those contours.
Set a piece of transparent paper over the drain. Mark around the shape of the drain with your pencil, expanding the perimeter by a quarter inch. Cut out the shape from the paper with your scissors.
Lay the paper circle on the piece of cardboard. Trace around it. Cut it out with the scissors. Save the circle.
Use your chalk snap line to divide the room into four squares, with two lines intersecting at the centre of the floor. (Note: If the drain is at the centre of the floor, readjust your lines so the intersection sits off one side of the drain, with at least a full tile's worth of space between the intersection and the edge of the drain.)
Apply thinset mortar over a few square feet of the floor, starting at the intersection of the lines. Press tiles into place along the lines, putting tile spacers between them. Lay all the full tiles that will fit, leaving clear a perimeter of floor around the drain.
Let the tiles set overnight. Remove the spacers.
Cut the and install the partial tiles near the walls, using your wet saw.
Set the last tiles over the drain without mortar, positioning them as if the drain wasn't there. Set the cardboard template on the tiles, lining it up with the drain underneath. Mark around the template onto the surface of each tile, using your pencil.
Cut the tiles along the pencil marks, using your wet saw. Spread mortar on the back of the cut tiles and set them in place by the drain, leaving a quarter inch between the cut tiles and the drain perimeter. Let the mortar set.
Spread grout over the whole floor with your grout trowel, forcing it into all the spaces, including the quarter-inch space between the cut edges of the tiles and the drain. Use a damp sponge to wipe up the excess grout.
Wear goggles when cutting tiles.