Ore generally doesn’t make its way to your inventory by itself. If you aren’t going to buy it, you will need to mine it. But where?
Typically, the higher the level of enemies in the zone, the higher quality ore you will find there.
A good rule of thumb is that you will find more ore in rocky and mountainous areas than you would in rolling plains. As such, the highest ore density is often found on the edges of a zone.
Mines and Caves often have much more veins than you’d normally find in outdoor areas. (and more enemies inside)
If you cannot seem to find any ore in a zone at all, it either contains few ore spawns, or is being excessively farmed. If it’s the latter, you might want to relocate to a different zone to improve your chances of finding ore.
Boosting Your Mining Skill
Your mining skill is naturally increased by mining, and a little by smelting at the very early levels. But you can boost your skill through an enchantment as well, allowing you to mine higher level ore where you wouldn’t otherwise be able to.
There are two enchantments:
Enchant Gloves – Mining: Permanently enchant gloves to give +2 mining skill.
It requires 1x Soul Dust and 3x Iron Ore.
Enchant Gloves – Advanced Mining: Permanently enchant gloves to give +5 mining skill.
It requires 3x Vision Dust and 3x Truesilver Bar.
Beside that, Goblin engineers can make a Goblin mining helmet (mail) for a further +5 boost.
I do recommend getting a pair of gloves enchanted with +5 mining, as it does help you mine ore deposits that would otherwise be just out of your reach.

Mining Locations Listed from best to worst, top to bottom.

Precious Metals These veins are typically rarer than the normal ore types