To begin, the Forge of Hephaistos can be found in the Hall of the Gods that is found after the prologue. The items that need to be upgraded include the sword, axe, bow, apollo's arrow, helmets, armors, and potion pouch. The weapons, helmets, and armors can be upgraded to 6 times each, while the arrows and potions need to be upgraded 8 times. Each upgrade costs a exponentially rising amount of adamantine shards. There are 4 types of shards which are yellow, blue, red, and purple.
Yellow - Arrows and Potions
Blue - All Weapons, Helmets, and Armor
Red - All Weapons
Purple - Helmets & Armor
All 4 types of shards can be found as crystals throughout the world that give a set amount of shards as a reward. Each type of shard can be found in different methods.
Yellow - Found in shard clusters and chests.
Blue - Found killing any enemy, shard clusters, chests, and even cutting shrubs or destroying vases. Update - Finish Are's questline and open up Ajax's fortress and keep spawning at the front gate and killing everyone there for a large far of blue shards.
Red - Found in shard clusters, killing unique enemies, and looting guarded chests.
Purple - Found in shard clusters, and looting epic chests.
I was able to complete this after about 35 in game hours and doing most of all the side content and main content in Valley of Eternal Spring, Grove of Kleos, Clashing Rocks, and The Forgelands.