I’ve made my first rig on a human model, but, as the title mentions, eyes, gum, teeth, and the tongue stay put in place since they are separate meshes. What am I missing? Do I have to parent them to the main mesh, and do the automatic weight armature again? Or else?
You should parent them to bones, not to main mesh.
Click on armature → edit mode (TAB) → click bone you want ot parent mesh to → object mode (TAB) → click on mesh → SHFT+ click on Armature → CTRL+P- > choose bone from menu