import bpy
import bmesh

obj = bpy.context.active_object
mesh = obj.data
bm = bmesh.new()
bm.from_mesh(mesh)

def element_id_layer(elems, id_name):
    """
    This function will create/print new layers
    """

    # Try to get layer of vertex IDs
    lay = bm.loops.layers.int.get(id_name)
    if lay is not None:
        print('Layer ', id_name, ' already exists')
    else:
        print('Creating layer ', id_name)
        lay = bm.loops.layers.int.new(id_name)
        # Set values in layer
        elems_iter = getattr(bm, elems)
        for elem in elems_iter:
            print(elem)
            if elems == 'faces':
                for loop in elem.loops:
                    loop[lay] = elem.index
                    print(loop[lay])
            else:
                for loop in elem.link_loops:
                    loop[lay] = elem.index
                    print(loop[lay])

element_id_layer('verts', 'VertIDs')
element_id_layer('edges', 'EdgeIDs')
element_id_layer('faces', 'FaceIDs')

bm.to_mesh(mesh)
bm.free()

Creating layer VertIDs
<BMVert(0x7fbb5da88010), index=0>
0
0
0
<BMVert(0x7fbb5da88048), index=1>
1
1
1
...

Creating layer  EdgeIDs
<BMEdge(0x7fbb5d942010), index=0, verts=(0x7fbb5da88010/0, 0x7fbb5da880f0/4)>
0
0
<BMEdge(0x7fbb5d942060), index=1, verts=(0x7fbb5da880f0/4, 0x7fbb5da88128/5)>
1
1
...

Creating layer  FaceIDs
<BMFace(0x7fbb5dafd010), index=0, totverts=4>
0
0
0
0
<BMFace(0x7fbb5dafd048), index=1, totverts=4>
1
1
1
1
...