Of course you cannot select the subdivided faces as long as the modifier is not applied, because the modifier is procedural and creates only "virtual" geometry, it is not real and can be changed anytime by changing the subdivision levels for example. You have to be careful with the terminology here to avoid misunderstandings: your title and question say, you have "applied" a modifier on the plane while you actually just added a modifier to it. "Applying" a modifier in Blender means, creating real geometry from the modifier's settings and removing the modifier, thus making the changes to the mesh permanent and no longer changeable.
Seeing the wireframe in Object Mode can be achieved in two ways:
- Per object by enabling Wireframe under Object Properties > Viewport Display.
- For all objects by enabling Wireframe in the Overlays for the 3D Viewport (the Opacity setting there effects the per object wireframe as well).
Note: If an object is set to Display As > Bounds in the Viewport Display options, it will only show the bounds and no wireframe, no matter where you enabled it (in Object Mode, but the mesh will be shown in Edit Mode).
Also an object with a Subdivision Surface modifier will not show the subdivided wireframe as long as Optimal Display is enabled in the modifier settings, no matter which method you choose.