This concept is so basic that I'll describe everything in the simplest terms, so that even a beginning beginner should be able to follow.
Fabric Scraps - you can even piece bits together, so you don't need much! Anything washable will do, but my first hint is that patterned fabric hides stains much better than solid colors.
Velcro - Just a small square of hook section and loop section. Do NOT use sticky velcro! (gums up the sewing machine and doesn't work without being sewn)
Paper - to draw out your pattern
Ruler & a couple of plates or something round - size dependent on the person you're sewing for
Sewing Machine - The one thing my Janome has that my beloved vintage machines don't have is a few decorative stitches - just enough to have a little fun with this project.
For the pattern, determine the width you want to cover the person's chest, and the length from shoulder to waist (or a bit longer or shorter, depending on your needs). My final pattern was a bit smaller than pictured below, since the Mumsie is pretty tiny. Once you've decided on your dimensions, find a large & a small circular object & trace your circles, as indicated below:
You will be cutting 2 pieces for each bib, and these will be sewn together, so allow room for seam allowances (1/4"-1/2") on all of the cut edges, including the small circle, which will be cut out for the neck.
Cut out your pieces, including the neck circle, and cut all the way through the strip above the neck circle. This will have a velcro attachment behind the neck. You can round off the corners of the strip ends if you want - this is optional.
I used 2 contrasting fabrics, and on this one, used the cut out circle as a decorative applique, sewing it on to the contrast side with a decorative stitch. If you're going to do anything like this, sew your applique on BEFORE sewing your two fabrics together.
Place your two fabrics with right sides together, matching all edges, pin, and sew all edges together, leaving about 2" free to turn the bib right side out. If you biggify this picture, you can see the 'free' section on the top left.
Reach through the section you left open & work the fabric through so that the right sides are now showing. Press the edges flat, trying to work the fabric out so that one side isn't rolled over the other at the edges. I use a large, sturdy pin to do this, and work it while I'm pressing. Don't worry if it isn't perfect - it's a bib, not a couture piece! ;-D
Now is the fun part! I don't even worry about the unsewn section that was left open for turning, because I just go around the entire piece with a decorative stitch. This adds a little sumpin' sumpin' to the design, and gives you a sturdy edge that won't roll around with use.
Stitch the velcro pieces on to the ends at the back of the neck section
And you're done!
Here's a sampling of the ones I made:
And a close up of the decorative stitches I used at the edges:
Quick, gratifying, practical, easily washable, and even fun to make!