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Easiest way to "sew" a quilt is to "tie" it. You take some embroidy floss threaded on a needle, poke it though all 3 layers of your quilt (from the top to the back, pull is through to the point you have a 3inch tail hanging out from the top, pop your needle back though (back to top), needle back though again, this time pull it tight, back up pulled tight, tie the "tail" to and the thread still on the needle in knot, tie another just to be safe. DONE!---I like to do them at corners so they don't take from the main flow of the quilt.
Sometimes I use buttons to give it an extra little pizzazz. But not for little kids quilts.
ETA----you need to "Baste" all 3 layers together before quilting so they don't all slip and slide as you quilt. You can do this several different ways as well. You can sew a "running" stitch in a GIANT X from one corner of your quilt to the other. Some folks add in a GIANT + from top to bottom and side to side just to be safe.
Me, I use either safety pins every 4inches or so. I stagger them like bricks every other row. Or I use what's called a "basting gun" It shoot those plastic thingies that hold tags on your clothes to hold the layers together.
The reason most folks sew the binding last is sometimes when you "baste" your quilt, it can be a little crooked and as you quilt you work your fabric out to the edges. Yes, you quilt from the middle out.