There are times when even the most dedicated machine sewer needs or wants to do some hand-work on a project, and it's useful to have a repertoire of stitches for different purposes. For the 20-minute vests, which can be sewn by hand quicker than you can thread your machine, these stitches will provide endless variations. For the seams, try a quilting thread, which is heavier than machine thread, so you need use only a single thread. If you want the stitches to be decorative, try cotton embroidery floss or a light weight knitting yarn. Use a short needle (easier for small stitches), with an eye large enough for the thread or yarn to move easily through.
Secure the thread with tiny back stitches at the beginning of the work and with a figure-8 knot at the end. For the figure-8 knot, take a tiny back stitch, wrap the thread under and around the needle before pulling the needle through. The backstitch makes a very strong seam. Take a 1/8" stitch, then insert the needle either 1/16 or 1/8" behind the thread, and take another 1/8" stitch.
The fell stitch is useful for joining two layers of fabric from the right side. Insert the needle directly below where it came out of the fabric, and bring it out 1/8" ahead and opposite where it first came out of the fabric. There will be diagonal stitches on the wrong side. The drawing stitch invisibly joins two layers of fabric with folded edges from the right side. Take a short stitch through one folded edge, then in the other folded edge.
The whipstitch is used for seaming fabrics, either right or wrong sides together. The stitches should be about 1/16" apart, and only as deep as necessary to create a firm seam. Leave a tail of thread when you start, and work several stitches over it to secure and hide the thread. The catch stitch is very useful for hemming as well as joining two edges. Take a small backstitch, move to the other layer, and take another backstitch.
The overcast stitch is used to prevent raveling. Work evenly, and as closely together as necessary for a secure edge. For a more secure edge, work a second row (crossed overcast stitches) back over the first rrow of overcast stitches.
The rolled hem uses a slipstitch to secure a very narrow finished edge. Roll the fabric between your thumb and finger (or, if the fabric won't roll, fold it), then secure it with a tiny stitch that catches just a thread or two from the fabric and then a short stitch through the roll (or the edge of the fabric). The baseball stitch is a decorative stitch to join two abutted edges. Insert the needle between the edges, bring it out 1/8" from one edge, insert between the edges again, and bring it out in the other edge. Work closely together for a secure seam.
The blanket stitch is a decorative stitch to finish either a raw edge. Work the stitches evenly, and as closely together as necessary to cover the raw edge. The buttonhole stitch controls fraying as well as creating a decorative edge. Put the needle under and through the fabric edge; loop the thread around the needle point, and pull the needle through. Tighten the thread, positioning the "knot" of the stitch at the fabric edge.

Sewing with fur, whether real or faux, is a bit different from sewing with fabric. When you cut the fabric, cut a single layer; cut only through the backing. Smooth the fur or pile away from the stitching line toward the fabric right side.

For hand sewing, use a whip stitch.

To sew by machine, use a zigzag stitch, with the "zig" going through both layers of fabric, and the "zag" falling just off the edge of the fabric. The seam should flatten out.

Most seams will be invisible from the right side if you tease the fur or pile out along the stitching line.