Farming and Laning are simple concepts that take time to master.

Farming grants you gold input by killing enemy minions. You try to score the last hit on a minion, the killing blow, as the killing blow decides who gets the gold. Keep in mind that about 10-15 minions are worth one champion kill, so if you have 30 minions and the enemy kills you once, you are still very likely ahead of them in gold.

Laning is exactly as implied: You go into a lane (top, mid, bot) and stay in that lane. The "laning phase" is the part of the game where everyone is focusing on farming minions and harassing their enemies. Some champions do better as farmers (such as most marksmen), gaining gold in the early game to increase their supreme late game power. Others do better as harassers (such as most assassins), attacking the enemy and stopping them from getting gold by farming. The laning phase ends when the first line of towers are destroyed and people start wandering the map for global objectives, such as dragon and baron, that give gold and buffs to the whole team.

Also, fyi, buffs are things that improve your stats, such as an armour buff improves your armour, and so on.