To parent your mesh to bones/armature: in Object Mode, select your Mesh FIRST then shift select the armature/bone you want to connect, right click over the mesh, choose "parent" and either "empty groups" or "auto groups".
After this, to go to weight paint: in Object Mode select armature/bone FIRST then your mesh then go to weight paint mode. Go to the "object data properties" tab (little green triangle) and you will see vertex groups. Select one and paint on it, this is applying weight to that vertex group.
To remove a section from a vertex group: Got to "Edit Mode" select the vertex group you want to remove something from, select the area on the mesh you want to remove from the vertex group by drag selecting. Over in the vertex group green triangle tab, click "remove" button below the vertex group list.