Some of them are hard to find. If you are alliance, the one in Goldshire is the glowing hole in the wall behind the anvil in the blacksmith shop - which for some odd reason doesn't say "forge" when you scroll over it. Blacksmith shops of this design also exist in Menethil Harbor, Darkshire, Theramore Island, Moonbrook, and many other places. Then there are smaller forges on stands of several designs - a human design in Stormwind City under a tent in the Dwarven District, one in Westfall, one in Thalanaar, one in Lakeshire, one on the upper level of Booty Bay, and many many more. And there is a Dwarf design, usually in a building spaced with anvils around a central fire found in Kharanos, Dun Modr, and other Dwarven areas. Both of those designs usually show up as "Forge" when you scroll over them.
Just keep your eyes open when you go into a town - if it has a blacksmith or weaponsmith, there is probably a forge nearby.
Unless you start as a Night Elf - as far as I know, there is no forge on their starting island or in Darnassus, and only one anvil. The nearest forge is in Auberdine on Darkshore, and unless you find a human or a dwarf around, you probably won't find a forge in an elven town.