Clean out the area along the front of the baseboard to remove any loose dirt or debris that will prevent a level installation of the trim. Use a vacuum to pull out the surface dirt and any particles that might stick in a crack between the floorboards.
Cut the back end off of the first strip of molding at a 45-degree angle with a miter saw. Set the leading end of the molding into a corner with a nail gun to anchor it to the floor or use a hammer and trim nails. Set the nails below the surface of the trim with a nailset and a hammer.
Add the second piece of molding but this time cut the leading edge at a 45-degree angle to match the first strip of molding's back end. Set each strip of molding in place, mark them with a pencil and cut as you work your way around the floor of the room. Nail the molding in place after you check to make sure the cuts fit well.
Use a mitered 45-degree cut for joining outside corners, a 45-degree beveled cut for joining runs of trim and a coping cut for inside corners, just as you would use for installing the baseboard.
Leave the quarter round unpainted to match the floorboards. This effect will visually mask the difference between the floor and the baseboard. Non-painted shoe molding also does not show the scuffs left from shoes or furniture hitting against it.