This is work in progress, I'll add pictures and more detailed info.
Don't hesitate to post feedback or precisions in the comments section!

I hope this will help you with mining in X Rebirth!

To start mining, you will need a ship that can carry bulk cargo and which has Surface Miner Drones.

The ship I have chosen is the Scaldis, which I have bought in a shipyard in the DeVries system.
It is one of the best cargo ships in my opinion, since it can carry all type of cargo AND help you with mining.

It costs quite a lot however, so you should consider bying a mining ship instead which I believe might be cheaper.

1. Make sure your ship is not in active mining mod

In order to check that, remotely call your ship, click "new order" and tell the ship to stop mining. If you don't have that option or the "new order" menu is greyed out, then it's OK.

Make sure the ship is in your squad.

2. Find a good mining spot

This is easy: just find an asteroid field.

3. Start mining

With the Skunk, switch to mining mode: menu ==> 1 (modes) ==> 3 (mining mode).

Select an asteroid with good yield (above 2000). Shoot it with the mining laser. It should explode in rather big bits. Select the big bit with the greatest yield. Shoot. Explode. Select the small bit with the greatest yield. Shoot again, explode again.

Now you should have tiny bits with an oval selection ring aroud them. When you hover over them, your cursor will turn into a small rock, and the text "Tag" appears under it.

All you have to do is double click the small piece of asteroid. A small missile will be launched towards it. It will then get a white outline.

After that, your mining ship will send the drones to pick up the ore/silicon/whatever you're mining.
That is basically it.

While mining, you can find loot container in the middle of the smallest asteroids.
These can contain rare minerals and gems that you can sell later to mining equipment dealers for good money.