Relics are leveled up by fusing other relics which costs Food. The food costs for each tier are as follows:

Bronze - 90 food per exp needed
Silver - 105 food per exp
Gold/Diamond/Obsidian - 120 food per exp

Fusing is similar to rune crafting. Keep in mind that the Relics you use in Fusion will permanently disappear.

You can fuse any relics together, and you are guaranteed to get relic experience points. The higher their tier, the more points you'll get. If you fuse the same Relics, you will get BONUS experience points.