Mining is a matching mini-game where you must match 3+ gems of a specific type and have to "mine" a certain number of each material to count as a "success". If you manage to get the total amount of all required materials, you get double of each material mined as a bonus. If unsuccessful, you still get to keep what you mined as a consolation prize.
1. Asteroids reset after you mine three other asteroids, successfully or unsuccessfully. Use this to your advantage!
2. Use the direction of matching to your advantage. Make the gems pull away from the side of the screen with the most materials, increasing your chance to match more later. You are also awarded more material for Nova and Supernova matches, which are more likely when matching this way.
3. There are the most asteroids on Beta Centauri (five total), which is close to Terra, although these can be challenging to mine successfully.
4. You need to complete (mine all required materials) from ten asteroids total. You can mine the same one multiple times, and the ones close to the beginning are the easiest. So just fly around the beginning areas and mine the easy ones until you get ten successes if you want this quickly.
EXTRA CREDIT: Don't forget to sell your extra materials for quick credits or save them for crafting items. You do have limited capacity to carry them depending on your ship. Some factions offer more for certain materials (price in green) and less for others (price in red).
(Let me know if anything is unclear instead of neg-voting and I will edit this as needed