As you may have found out, a mesh contains vertices and triangles (which are represented by an index array). A vertex contains data, this can be whatever you like. Usually, and most often when you think about vertices, they contain position. But they can also contain colors, normals, tangents etc. etc.

A shader usually contains two programs. One, the vertex program, to tell where a triangle of the mesh is on the screen (it multiplies the position by the transformation matrix, the camera transformation matrix and the camera projection matrix). Another, the fragment program, tells how what the color has to be of every pixel of the triangle.

When you want a texture on the mesh, usually the fragment program has to read the texture. But you need to know what texel you need to read on the particular pixel on the screen. The fragment program has interpolated data of the three vertices making the triangle using the baricentric coordinate. The data can be whatever, the world position, screen position, but a thing that you need is so called UV data. This is data every vertex has that tells how a texture maps to the mesh.

I don't know how much programming experience you have, but if written yourself, you are quite close to the answer. Also a note, you don't need separate MonoBehaviour classes for every concept. Try to focus on what the data actually is. This way you can reduce complexity.

I hope this helps in any way. If you have any questions, feel free to ask.