In Final Fantasy II, there is no way to really "level up". Instead, there is a realistic stat system that makes you stronger in certain areas based on how you fight. There is an easy way to manipulate magic and weapon levels in this game.

When you enter a battle, select the spell (or, if you wish to strengthen your weapon proficiency, select attack) you wish to level up and confirm your target(s). When your next character comes up, press B to go back to the person you are working with. Continue this as long as you wish to easily gain a level with your magic and/or weapons. Note that you can only level up each spell and weapon skill one time in every battle.