How do I make just the neck move and have his shirt stay in place? An example is in the first picture. In the second photo it shows the layers I have in Illustrator. Am I supposed to put the + sign next to name on the neck layer?
There are two strategies with different details on how to get it fully working.
The first approach is to put some pins along the bottom of the character and maybe a stick across the top of the shoulders. The pins stop that part moving relative to the parent (the scene in your case), the sticks stop/reduce bending of the artwork forcing other areas to bend (the neck being your goal).
The other approach is using independence, which the "+" character turns on. Then the artwork will rotate at the attachment point (the origin) of the head. You may need to draw the neck up taller behind the head to not reveal strange looking gaps if you do it this way. The head and shoulders will not move however as the head is 'independent' to the body, so the pins and sticks etc are not needed.
Both ways are valid - it just depends what look you prefer. As a general rule, I find longer necks have less need to be independent.