You might have heard of this, or consider it cheating but what the hey. Here's a great method to power level right at the start of the game.
When you first control Vaan around Rabanastre, after the parade and you have access to the Westersands, travel to the Corridor of Sands through the first exit to the north when you leave Rabanastre. Before you leave for the Westersands, make sure you've got some Phoenix Downs.
Stay near the zone line, don't disturb the Cockatrice rolling about a few yards ahead of you as it's highly unlikely you'll beat it at Lvl 1 or 2.
Go to the config menu, set the battle speed to slow, and the mode to active. Then, beat yourself up until your health falls below 10%.
The rare game Dustia will appear, as it's undead, QUICKLY use a Phoenix Down on it to kill it instantly and gain 1100 exp. You can gain around 20 levels in an hour. Make sure you're quick. This is why you set the battle mode to active, so you can quickly move to select a Phoenix Down from your item menu as Dustia is not targetable until a few seconds after the white haze appears. Also, the slower the battle speed is, the longer he will take to prepare his Dark spell, and as the usage of an item is instant, you should be able to kill him off before he uses it. If you're too slow and he manages to hit you with his spell, it's game over.
Also, another word of advice... try not to accidentally target the Cockatrice! Lather, rinse and repeat but make sure you save every few levels in case Dustia manages to get the better of you...
It's best to use this method before you join with Penelo for the first time at the Nomad Village, as she'll be at the level you're on when she joins you.
Enjoy =]
Edit:
Also, forgot to add that you need to zone out twice to get Dustia to appear again so it's worth going back to Rabanastre and saving after every time you fight it anyway.