Appliqueing a onesie

I am having a lot of trouble with the stitching around the applique.  No matter what I try, either my thread doesn't catch, my thread rips, or the onesie get's pulled into the bobbin.  Any advice?

Are you using a stabilizer on the back of where you are appliqueing?  This could help with issues of the material being pulled into the hole in your machine.  The stabilizer will help make the garment stiffer as it is guided through your machine.

Or sew it by hand with some embroidery floss. 

what did you use to attach your applique to the onesie itself...it really helps with this if you have attached it using some kind of fusible web. then you HAVE to stabilize the onesie or you are just asking for headaches. I use sulky iron-on/tear off for this and works great. Watch for the length of your stitch too if it is too tight that might make make it a little tougher too. I like to use a medium sized zig zag. Good luck...

I always use a double-sided iron-on adhesive and favor Heat-n-Bond lite for onesies and t-shirts.  It adds the extra stabilizer you need to sew without anything getting stuck in the hole.  The adhesive has a paper side which I draw the appliqué design onto, then iron it onto the fabric, cut out, peel paper off, iron it onto onesie and sew.  I've made dozens of shirts now which have been washed and worn alot without problems.

You may also check your tension and be sure its not too high.  From my own experience, I've noticed that the thicker the fabric, the higher the tension number.  I usually have mine set at 2-3 for onesies. 

Lastly, I don't try to do any intricate designs onto onesies.  For something like a monkey or horse, I will sew the appliqué onto a square "patch" and then sew the simple square onto the onesie.  The thin knit of the onesie will not handle multiple turns or heavy stitches well. 

Keep trying.  I love an appliquéd onesie/shirt.  It's a quick and easy way to add a special touch.   

when you are appliqueing letters do you always sew around it? does the fusible web hold it through laundering? I am having so much trouble sewing around letters with the small holes. I am doing this on regular cotton though, not a onesie.

I also use hand embroidery floss to affix to my onesies... good luck, lots of fun

I skip appliqueing around small letter holes.  The heat-n-bond I use will usually do a good job of holding it in place and if it frays a little, I usually don't mind the look. 
You could also not stitch around the entire hole if it's too hard, but just the top half or bottom half which will give you a better chance of the applique staying down.

I recently did some applique work on a flannel baby blanket.  Since it was the middle of the night (lol) and I didn't have any fusible webbing of any sort, I was stuck for ideas...until I glanced over and saw my glue stick sitting on my crafting table.  I glued the shape (a whale out of denim) onto the flannel and that kept it from moving around.  Because I was using denim, it was stiff enough that I didn't need to use a stabilizer, but I had a back up plan for that too, a left over piece of fabric that had interfacing already attached. 

Make sure you're using something to hold the applique on (which it doesn't sound like you're having an issue with) and use something stiff inside the onesie to back the applique with...there is a ton of great advice in the other responses, hope you have success!

when doing any machine embrodery work on material it is necessary to use a stablizer. on infant garments i use a wash away stablizer so that nothing iritates the babies skin. stablizer holds the fabric so that it does not lose its shape. after completing the garment the stablizer can be washed out. i find that i will do the embrodery work on the fabric before i construct the garment. i hope that this helps with your project. if you use the correct stablizer it will come out a beautiful project.