Tiling around a round sink can be complicated. It will be less complicated with a self-rimming sink---a sink featuring a rim that extends out over the sides of the bowl and sits on the tile. With a self-rimming sink, you won't have to finish the edge between the tile and sink, because it will be covered. The hole for your self-rimming sink should be slightly larger than the bowl of the sink, and the rim of the sink should extend for an inch or two outside the hole.
Install your tile around the perimeter of the sink hole. Get the tile as close to the sink hole as you can without making cuts. Let them set. Don't grout yet.
Lay a piece of tracing paper over the sink hole. Mark around the edges of it to get the perimeter of the sink. Cut out the circle with your scissors.
Set the tracing paper on your cardboard, mark around the edges of it, and cut it out. You now have a cardboard template the size of the sink hole.
Lay tiles loosely along the edge of the hole for the sink, spaced correctly with the previously set tiles.
Set your cardboard template on top of the tiles, lined up with the sink hole below. You'll be able to see the perimeter of the hole through the spaces between the tiles. With your pencil, mark the perimeter of the template on each tile.
Cut each tile on your wet saw.
Spread adhesive around the untiled sink area. Set the cut tiles in place. Let them dry overnight.
Grout all the tiles at once, pressing the grout into the lines with your grout float. Wipe the excess off the tile face with a damp sponge. Let the grout set for two days.
Set your self-rimming sink in place, with the rim edge covering the tiles. Caulk around the sink.
Wear eye protection when cutting your tiles.