How can I get Ingots and Cores?

Ingots and Cores are necessary to upgrade your items.
You can find Ingots when winning battles, buy them at different merchants, and as part of any ship cargo:

Cores are slightly harder to come by. Get them by winning battles against bosses, from the Nordvig Merchant, and as part of the Smuggler or Imperial Ship cargo:

Anthracite Cores can be obtained in the Loot Wagons of the Invaders:
Reach the Warlords Level 23 as well as the level 50 of the Brigand Island for Invaders to start popping up on your map.

If you find yourself missing Ingots of certain rarities, you can make use of the 'Combine' and 'Split' functionalities in the Forge. Combining allows you to merge several lower rarity Ingots into one higher quality Ingot, while splitting works the other way round: you can break higher rarities Ingots apart to get several lower rarity Ingots:
To upgrade items to levels higher than 61, you need to use Anthracite, rather than Ingots. Anthracite works much like Ingots, but it cannot be split or merged. Get Anthracite and Sacred Cores in battles against Invaders. Invaders will start showing up on your map once you have reached the Warlord Level 23, and unlocked level 50 in the Brigand Island.