Display wireframe and solid color

To do that, a possibility is to use a GLSL fragment shader that changes the fragment color when the fragment is near one edge of the triangle. Here is the GLSL shader that I am using. As input, it takes the barycentric coordinates of the fragment in the triangle, and an edge mask that selects for each edge whether it should be drawn or not. (rem: I had to use it with the compatibility profile for backward compatibility reasons, if you do not want to do that, it can easily be adapted):

(fragment source)

#version 150 compatibility

flat in float diffuse;
flat in float specular;
flat in vec3  edge_mask;
in vec2 bary;
uniform float mesh_width = 1.0;
uniform vec3 mesh_color = vec3(0.0, 0.0, 0.0);
uniform bool lighting = true;
out vec4 frag_color;

float edge_factor(){
    vec3 bary3 = vec3(bary.x, bary.y, 1.0-bary.x-bary.y);
    vec3 d = fwidth(bary3);
    vec3 a3 = smoothstep(vec3(0.0,0.0,0.0), d*mesh_width, bary3);
    a3 = vec3(1.0, 1.0, 1.0) - edge_mask + edge_mask*a3;
    return min(min(a3.x, a3.y), a3.z);
}

void main() {
    float s = (lighting && gl_FrontFacing) ? 1.0 : -1.0;
    vec4  Kdiff = gl_FrontFacing ?
         gl_FrontMaterial.diffuse : gl_BackMaterial.diffuse;
    float sdiffuse = s * diffuse;
    vec4 result = vec4(0.1, 0.1, 0.1, 1.0);
    if(sdiffuse > 0.0) {
       result += sdiffuse*Kdiff +
                 specular*gl_FrontMaterial.specular;
    }
    frag_color = (mesh_width != 0.0) ?
                  mix(vec4(mesh_color,1.0),result,edge_factor()) :
                  result;
}

To render both a model and its wireframe, you can use a pattern like this one:

// mesh
var material = new THREE.MeshPhongMaterial( {
    color: 0xff0000,
    polygonOffset: true,
    polygonOffsetFactor: 1, // positive value pushes polygon further away
    polygonOffsetUnits: 1
} );
var mesh = new THREE.Mesh( geometry, material );
scene.add( mesh )

// wireframe
var geo = new THREE.EdgesGeometry( mesh.geometry ); // or WireframeGeometry
var mat = new THREE.LineBasicMaterial( { color: 0xffffff } );
var wireframe = new THREE.LineSegments( geo, mat );
mesh.add( wireframe );

The use of polygonOffset will help prevent z-fighting between the mesh material and the wireframe line. Consequently, the wireframe will look a lot better.

three.js r.126

This can also be achieved with WireframeGeometry: https://threejs.org/docs/#api/en/geometries/WireframeGeometry. (and give plane and line the same position, you can also play with opacity see the docs).

let geometryWithFillAndWireFrame = () => {

    let geometry = new THREE.PlaneGeometry(250, 250, 10, 10);
    let material = new THREE.MeshBasicMaterial( { color: 0xd3d3d3} );
    let plane = new THREE.Mesh(geometry, material);

    scene.add(plane);

    let wireframe = new THREE.WireframeGeometry( geometry );

    let line = new THREE.LineSegments( wireframe );
        
    line.material.color.setHex(0x000000);
        
    scene.add(line);
        
};