Barbara,
If you are going to machine applique you either have to use a satin stitch which is a zig zag very close together or stabilize the edge somehow. You could iron the raw edges under before you stitch or if you use fuseable. Fuse a bigger piece of fabric than you need and then cut it down to the right size. That way the glue will go all the way to the edge. Raw edge applique is usually only used for wallhangings or things that you want to fray.
I save used dryer sheets and make my appliques have turned edges by stitching the fabric right side together with the dryer sheet. By sewing the two together on what was the cutting line I now have the perfect shapped applique. Then I can cut a slit in the dryer sheet and turn the applique to the right side. The dryer sheet is thin enough to not add much bulk and it keeps the raw edges turned under. I can then stitch the appliques to the quilt by any method. Hand or machine.