Short version: Your mesh would follow your bones in Blender if your mesh got some armature modifier with a matching bone = vertex-group name and some weights on it.
Long version - you would need:
You need a vertex group with the same name as your bone name.
You need this vertex groups with matching names for every mesh (object) that should follow that bone (body, cloth, hair, …).
You could find the vertex group box if you select your mesh objects (in object mode) and click on the triangle symbol in the properties tab.
You could create a vertex group manually or get a full list of group names if you select your mesh and afterwards your armature and “Set Parent To” → “Armature Deform” “With Automatic Weights” or “With Empty Groups”
If you would not use the Mirror modifier (anymore) but still would use the “X Mirror” option in weightpaint mode (for example if your UVmap is already finished) then make sure your bone/vertex-group-names ends with “L” and “R” like “upperarm_L” and “upperarm_R”.
You could get some weights (more or less working) via automatic weights with the already mentioned parenting “With Automatic Weights”
You could get some weights if you select some faces or vertices in edit mode and click on your “Assign” button in the vertex group box in your properties tab.
You could get some weights if you select the vertex group (or bone) in weight paint mode and paint on your mesh.
If you would do some weight painting then I would recommend to select your armature and switch to the pose mode, select the “human” (armature) symbol in your properties tab and enable “X ray” in the Display pulldown. Afterwards right-click on your mesh in the 3D view (which should select it) and from there switch to weight-paint. That way you could directly right-click your bones in the 3D view of the weight-paint mode (which makes weight-painting a lot easier).
You don’t need the armature as parent but it is handy so I would keep it. It’s not the parent part but the armature modifier that is responsible that your mesh is following your bones in Blender.