I'm sure people will have different advice, but I just finished three embroidered onsies. Here's the link so you check out the results of my advice http://www.craftster.org/forum/index.php?topic=176710.msg1827717;topicseen#msg1827717
I used fabric glue to applique a patch of fabric that i embroidered the edges of, so that was quite stiff, but for the words under each picture, I just stitched them right on--no stabilizer or anything.
I did use a hoop, but I didn't pull it super tight. I mostly used it to keep the extra fabric out of the way and to help me stitch straight. I also made sure not to pull my stitches really tight.
I know other people on here use a stabilizer, but I just went for it.
Hope that helps a bit!