Continue playing through the very early missions in multiplayer and soon Cid's crafting bench will unlock directly across from where you receive your missions from. Simply go over to it and select the weapon you wish to upgrade. Make sure its a weapon that caps out at level 30 not 20, otherwise it won't work. Now all the scrap and treasure you've been collecting by defeating monsters can be used to power up your weapons. Select the amount you wish to spend and the bottom right corner will show you how much of an upgrade you will receive for those items. Note that each item offers different EXP depending on rarity.
Very simple and can easily be obtained by using common junk to craft.