Copper Ore is the lowest level ore that you can farm. It comes from Copper Veins and mostly found at the starting areas and in lower level zones.
There is not much difference between the zones where you can farm Copper Ore, but the zones listed below are slightly better than the other alternatives. If you are below level 20, you should stay at the zone where you are questing currently, there will be plenty of Copper Ore there.
Durotar - Horde
Durotar is an excellent place to farm Copper Ore, there are a few caves in this zone and usually there are 3-4 Copper Veins inside those caves.
Darkshore - Alliance
Darkshore is the best place for Alliance players. Even Horde players can go there with flying mount. I was able to get 200 Copper Ore with just one lap in this zone.
IMPORTANT! If you are level 110, you will see an updated map of Darkshore. But, if you talk to Zidormi, she will allow you to switch between the new and old phases of Darkshore. (if you open the map, she will be marked with a white text bubble at north)
These are the best zones if you want to farm Copper. Once you are done farming Copper Ore, go to my Tin Ore Farming page.