In contrast to leveling up your characters, leveling up the Keyblades is not as simple as just defeating enemies to get experience points. Instead you must upgrade them from a special menu that is found in the Moogle Shop. So you can only level up your Keyblades when you are at a Moogle Shop.
Anyway, when you are at a Moogle Shop, click on the "Keyblade Forge" submenu to open up the menu where you can upgrade your Keyblades. Each level requires a certain set of materials, and all Keyblades require exactly the same things. However, each higher level will require more materials than before, so it takes a lot materials to get a Keyblade from level 8 to 9 than it does to get it from level 1 to level 2.
If you have the materials ready, simply press A or X depending on your console to confirm that you want to level up the Keyblade, and it's done. Nothing else is required!
Each Keyblade can be level up to level 10, and they all start at level 0.
The materials needed for each level of the KeybladesBelow is the full list of all the items you need for leveling up each level of the Keyblade.
Level 1: Fluorite x1, Wellspring Shard x2
Level 2: Fluorite x1, Wellspring Shard x3
Level 3: Fluorite x1, Wellspring Shard x4
Level 4: Damascus x1, Wellspring Stone x1
Level 5: Damascus x1, Wellspring Stone x1, Pulsing Stone x1
Level 6: Damascus x1, Wellspring Stone x3
Level 7: Adamantite x1, Wellspring Gem x1
Level 8: Adamantite x1, Wellspring Gem x2
Level 9: Adamantite x1, Wellspring Gem x3
Level 10: Adamantite x2, Wellspring Gem x3