I want it to look like the top image when I press P, so it has those lines around each face.
Not like this:
You want to create a wireframe material. A quick 'n' dirty way to do generate a wireframe texture is described below:
Unwrap your model. If you don't know how to do this simply select everything and press CtrlE > Mark Seam.
Press U > Unwrap.
Open up a UV/Image editor:
Click UVs > Export UV Layout. Remember where you save this image.
Press AltO to open a texture, then select the saved image.
You may want to remove the transparency in an image editor if you want your model to be opaque.