For this tutorial I use a serger, which is what makes this so, so simple to do! First I found a sweater that I did not want any longer, didn't fit me right, etc. but that I still "liked," and wasn't too bulky.
Then I shortened the arms by cutting them off. Just like this here. Measure you child's arm length to get the right length for your sweater recycling project. You can use the sleeve remnants for arm or leg warmers! Cut the length of the sweater as needed as well.
In these photos you can see that I have serged the back seam. After it is serged I lay the back flat and flatten the serged seam; I run a straight stitch all the way down the back so that it looks clean and tidy. (the left photo shows cutting off the excess with the serger. the right photo shows flattening the seam and sewing a straight stitch all the way down.)
Here is what the sweater looks like after you are finished with the back seam.
Then go ahead and serge the arm sleeves and the bottom of your sweater if you took off length there.
To add an applique to your sweater it is also really simple to do: First you will need to find your image or shape. I chose Godzilla for this one. Trace it on to a piece of heat n' bond (the paper side; that won't be being applied to your fabric).Then iron it on to the WRONG side of your fabric piece, follow manufacturers instructions. You will peel the backing away, as shown, and place it on your sweater; reheat it with an iron to affix it in place.
After it is on your sweater you will then sew around your applique to keep it secure through wear and washings. I often use a small zigzag stitch.You can also add a button or any other embellishments you may want to add at this time. That's it you are done! It's really simple and a great way to upcycle/recycle old or unused cardigan sweaters.
Here's another one I did using a cropped sleeved sweater cardigan.