Dear Anna,
First you need to find simple drawings of the animals you want to use. Try childrens' story or coloring books. I would do applique. It is not stiff if you use steam-a-seam lite. Then you can do raw edge applique with the buttonhole stitch on your machine. Also to avoid a stiff baby quilt do not over-do the quilting. I have several quilts I made 35 yrs. ago, when the kids were small that are just fine today. The quilting lines are 2" or 3" apart.