I do what Ginger and Annette do except that I use fabric (and sometimes batting, too) rather than a fusible web.

I use steam a seam lite and then just straight stitch around the pueces with a smaller stitch.

Fusible webbing is a quilter's best friend when applique is involved. Wonder Under is a fantastic product. I use a small zig zag stitch, satin stitch, buttonhole stitch on my applique and so far, knock wood, it has proven to be the right choice. Also if you use wool felt for your pieces, something not always the best choice, then you have no worries about frayed edges. Take a look at Becky Goldsmith's "Applique The Piece o' Cake Way" as it is an excellent guide.

I also use Steam a Seam Lite and then either satin stitch or blanket stitch around the edges, depending on the look I want.

Watson

I use a light weight fusible garment interfacing for machine applique. Here's a video that shows how: https://www.youtube.com/watch?v=8ChE9UBWA8A
Also check out Eleanor Burns machine applique method too.

As long as you use a fairly small stitch, there is minimum fraying with a blanket stitch and fused appliqué. Also, partly it depends how straight the appliqué is. Curvy appliqué frays less because there aren't long lengths of the warp/weft that can come unraveled at once.

I do not bother with turning the edges under. I can't take that long to do it and besides, my hands get too sore and achy to do so. So I cut out my appliques, use Steam a Seam, and then satin stitch around the edges, or even a simple blanket stitch for wall hangings.

You can also raw edge appliqué were you use a fusible and then use basically a straight free motion quilting Seam and you go around 3 to 4 times. I like using a slightly darker thread. Generally you might have a little fraying if you don't get exactly on the edge but for baby quilts depending on what your appliquing it can be a cute feature.

I used to satin stitch, but did not personally like the bulk.