There are 4 major places that one can level up through out the whole game, 2 in The World of Light and 3 in The World of Ruin. The first one is the Lete River. It works best if you have an Auto controller and if you put the game on Memory (it's in the Game Control Screen). What you do is while on the Lete River, you will reach a place where it ask's you to go either Up or Left. Choose whichever one it is already on and go (I think that it is Up, but I'm not sure). Then put your controller on auto-fire and also make sure that Banon is on Heal and everyone else is on attack, and that the Game is on Memory or this will not work. Also make sure that Banon is in the back row so that he takes less damage. Leave your game on for up to 6 days checking it every once in a while to make sure it is still working. I also suggest you turn your T.V. off, not the game just the T.V. so that it doesn't overheat. If you want, you can save by killing Banon. and you you will go back to you're last save point, and you will still have all of you're levels. Save then go back and start leveling up again.
The Second one is on the Floating Isle at the end of the World of Light. The Monsters there give good experience for that time in the game. So before you get into the big fight with Atma, you might want to stick around there for a little while and level up, and jump off the Isle whenever you run out of life or supplies.
In the World of Ruin the most popular place for me to level up was in the Forest just North of the Veldt. In there you will find Tyranosaurs and Brachiosaurs (which can randomly drop Economizers). Usually you will fight Tyranosaurs in either groups of 2 or just one of them. Their normal attack is powerfull so watch out, and they have the power to cast Meteor which will do up to 2000 pts. of damage to each person. But after awhile there you will be happy that you spent the time there.
The Second place in the World of Ruin doesn't level up your Experience as much as your Magic points. In the Desert just East of the Opera House, you will fight 2 different enemies: Hoover and Cactus. Hoover which is a pain to fight because all he usually does is cast Sandstorm throughout the fight. It's a spell that does between 1000 and 1500 pts. of damage. But when you defeat him you will get 5 magic points. As for Cactus, he gives you 10 magic points, but you have to kill him quick or he'll run away. He only has 3 life points, but very few weapons can hit him. If you have the Genjie Glove and Offering on one of your characters you will be able to hit him and kill him, also the Illumna can also hit him for 1 I believe.
The last good place to level up would of course be Kefkas Tower. Right be for you fight Kefka, if you stay there and fight a few times you will find that the enemies are giving you good experience, and that you can level up a few times before the final fight.