Try a wash-away stabilizer. Some of this just depends on the design. Some are dense, some are more open. Applique designs work well for baby stuff...try DesignsbyJuJu. She runs awesome specials, the designs stitch out beautifully, and there are tons of child oriented designs, in fact, that's what most of her designs are. A little experimentation goes a long way as you learn to read designs. Keep an eye on the number of stitches in the design. 35,000 stitches in a 5 x 5 design is going to be dense.
By the way, I use one layer of tear-away for her designs.
And I'm not affiliated, just a happy customer!