Steam a Seam 2 is a fabric adhesive that won't gunk up your needle. There are a several different things you can use. You can probably find a turorial on line of several different ways to do it.

Are you going to stitch around the applique? There is a way to do it using fusible interfacing, where you sew the fabric and the interfacing, around the shape, or in a square or a circle, then cut a slit in the interfacing, and turn it inside out so the bumpy glue side is facing out on one side, and your picture is right side up on the other. Then you can fuse it to your t-shirt so you don't have any raw edges, then put a finishing stitch around your picture. I have not personally tried this, as I am still a newbie, but read it in several books.

If you use fusible interfacing, be sure that it can be washed and dried in the same manner as the t-shirt - so if the shirt will go in the dryer with high heat, be sure the interfacing can do the same. Guess how I learned about that!!

As for the needle, use a ball point needle or a stretch needle. A size 80 should be fine.

If you haven't already finished your project, here's a solution. You can buy paper backed adhesive that you cut to size and iron onto the back of the picture. You peel the paper off and iron the picture on top of the other t-shirt.

Finishing around the border of a applique can be difficult on knits, depending on the fabric. I haven't used it, but the link below mentions no sew fusible webbing.

Here is a link that might be useful: Fusible applique

'Wonder Under' is 1 brand name of paper-backed fusible - great stuff for appliques = )