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Blacksmithing is the ability to craft items and materials out of scrap metal.
At first, you won't even be able to use a forge to level up. Completing the blacksmithing quest will be the only way to gain XP for the first blacksmithing level. The quest is found at the survivors' camp near Tula (southwest of Moscow) where you are given a quest by a survivor named Ignat. Once finished, you can go to Ignat's "eternal" furnace to begin.
Note: Ignat's furnace is "eternal" because it doesn't extinguish. Sadly, you can't build a workbench there.
Once you can craft bellows and build a forge, the most efficient way to level up is to craft items. Iron pipes are strongly recommended, as they provide 10 Xp per bellows use, and they are needed in multiple crafting recipes.