I use a lightweight iron on and do not hoop the onesie. Just to provide a base for stitching, I will use a Sticky Stabilizer which I hoop and then "stick the area to be sewn on top of the hoop. I use a water soluuable on top which keeps the stitches on top of the knit. After stitching, I use something like "Cover All" a thin soft stabilizer which irons on over the back stitching to keep the design from irritating baby's skin. Also, use a Ball Poiint Needle rather than a sharp. The ballpoint goes between the fibers and doesn't piece it like a Sharp.
Using just straight stabilizer such as tearaway or cutaway will not hold the woven knit in place and although it might look good prior to washing, will usually pucker after washing.
I also do all my t-shirts in this same method although I might not use the Cover All on the back depending on the age of the child.
Embroidery Library is my "go to" site when I have questions regarding stabilizers/thread/fabrics. I printed out the recommendations list and put it in a binder for easy reference.