In order to upgrade a weapon, you must first pick up an issue of 'Weapons Monthly'. The first issue is an unmissable drop from the Elvoret boss atop the Dollet Communications Tower during the SeeD field exam mission. However, this issue's upgrades are for the base weapons you already have (so you can downgrade them later for whatever reason), so in reality the first issue you should pick up will be on the bedside table in Squall's new dormitory once you've attended the SeeD ball; It should be easy to spot as it stands out from the background. The magazine unlocks these upgrades:
Valiant (Irvine): 12 STR, 105 HIT / Requires 4x Screw, 1x Steel Pipe
Maverick (Zell): 15 STR, 99 HIT / Requires 1x Dragon Fin, 1x Spider Web
Pinwheel (Rinoa): 11 STR, 99 HIT / Requires 3x M-Stone Piece
Shear Trigger (Squall): 14 STR, 225 HIT / Requires 4x Screw, 1x Steel Pipe
The Pinwheel and Valiant are both Rinoa and Irvine's base weapons respectively, so the first real options are Zell's Maverick and Squall's Shear Trigger, and Squall's weapon is the easier/quicker choice. The first chance you'll get to gather the resources for Shear Trigger is after you reach the area around Deling City. When running on the brown patches of land you may encounter 'Geezard' enemies, which are common and drop screws, of which we need 4. You will also encounter 'Wendigo' enemies, which are also pretty common and have a chance to drop steel pipes.
Once you have 4 screws and 1 steel pipe, head back to Deling City and make your way through the maze of screens to the high street (if you hit the hotel, go down one screen). The weapon upgrade shop is on the right hand side of the street, and is the door closer to the screen. Head inside, choose to upgrade to Sheer Trigger, and your achievement should unlock. No other upgrades are necessary, but any of them will work, so if you find yourself in the endgame, you've picked up magazines, and you've fought a bunch of enemies, chances are you'll have the components for something.