What a shame to see such a cute quilt come apart like that; I'm so sorry this happened!
I try to make my baby quilts as indestrucable as possible, so some of the things I pay special attention to include:
Start out with good quality quilting cotton with a nice tight weave.
Prewash, starch and press all the fabric before cutting.
Choose a good quality thread both for piecing and quilting.
Full 1/4 inch seams.
Shorter than "normal" stitch length.
Pressing the seams all to one side as I construct the top.
Machine quilting (usually a combo of FMQ and stich in the ditch), no farther apart than recommened for the batting I am using.
I didn't learn this all at once either, and totally understand how you must feel. I've had some mishaps as I go along too, and just try to learn from them.