If I were hand quilting, when I came to the applique I would carry my thread between the layers to my next point to start quilting; however, since I quilt with a longarm machine I would quilt up to the applique, stitch along side of the applique to the point of my next crosshatch line and begin stitching again. I would match my thread to the background fabric color. If the backing is a printed fabric wonderful(!) but if it is a solid color I just hope for a beautiful stitched design to show. When I do need to stop a stitching line I either take tiny stitches, backstitch two or three stitches, or tie off and bury the ends (weaving the ends thru the batting and in a direction that my next stitching will stitch over them to help secure the ends.