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In the very beginning your first ship already has cheap weapons and the Civilian Shield Booster on board, obtained after the training course. This is already enough to prey on weak pirates.
After the training and the first missions the player will already have about 15,000 ISK. You need to spend some of that money on a cheap weapons. Get as many weapons as well as ammunition as you can. Try to buy ammo with large radius of destruction, and low levels of minus modifier radius, because if you miss the target from a distance then damage rate is unimportant.
For the novice hunter for pirates it's better to go to the 0.8 sectors first. Find there a system with multiple asteroid belts and patrol them. Most of the pirates from there are weak and travel alone.
Keep the auto-scanner on, and when you see a pirate, right click on it to lock the target. Pirates can be recognized by the "+" sign, do not confuse them with neutral NPCs - Concord or merchant ships!
While attacking the enemy, use the option "orbit - X km", to be always on a distance from the enemy and shoot him, orbiting him. "X" is the optimal radius to shoot with your arms.
In many cases, especially in the solar systems with a high level of security, you can shoot from a greater range than NPC pirates can. If you're lucky, you'll need just a few shots to kill a pirate, and you yourself will be safe.
Then pick up the container with the cargo. You will find something that you can use in it, which you'll be able to sell or recycle. This will be your first kill in the game, and the first prize! For the head of a killed pirate you usually receive 1000-1500 ISK, but nevertheless it's money, plus the loot, and all together it's much more fascinating than mining.