Hi BlackbirdM3. To get the movement out you have to remove the stem and crown together. I don't have a self-winding movement to show you as an example, so I'm going to use this 10AE movement.
You need to loosen the little screw just to the left under the crown about two turns. Then gently pull on the crown and see if the stem will come out. If not, loosen it about a 1/2 turn at a time and try pulling the stem out. Do Not Completely back this screw out or you will have to take the hands and face off the watch to put it back in.
When you go to put it back in, push the stem in until it's seated and snug this screw up. Then pull the stem out to the set position and tighten the screw down. Don't use excessive force or you will break the screw. If the stem and crown pull completely out when you snug the screw down you didn't have it seated completely. Push it back in and slowly loosen this screw until you feel the stem seat and then snug it down as mentioned above.