A character's Level is determined by how many experience points that character has. The higher your level, the higher all your other Statistics will be. In any Golden Sun game, the highest level any character can reach is Lv. 99.
In Golden Sun the starting level for Isaac, Garet, and Jenna is Lv.1. When Ivan joins Isaac's party he is at Lv.4, and later when Mia joins she is at Lv.10. In Golden Sun: The Lost Age the starting level for Felix, Jenna, and Sheba is Lv.5; later, Piers joins when he is at Lv.18. If players choose to use a password enhanced file in The Lost Age, Isaac, Garet, Ivan, and Mia will all be the level at which they left off in the original Golden Sun. If players choose to use Easy Mode in The Lost Age all the Adepts will start the game at the level that were when the game was completed. In Golden Sun: Dark Dawn, the party members all begin at various levels: Matthew and Karis at Lv. 1, Tyrell at Lv. 4, Rief at Lv. 10, Amiti at Lv. 14, Sveta at level 22, Eoleo at Lv. 32, and Himi at Lv. 35.
The only way to increase your characters' level is by engaging in battles with monsters. Typically bosses yield more experience points than others, but since bosses are much more powerful than normal monsters and are generally only encountered once, fighting bosses is not a feasible way to train.
After each battle, you will be rewarded with experience points. Each character that participated in the battle will get the full amount, whereas characters that were not involved get half as many.
In Golden Sun
The most effective way of training in the original Golden Sun is to repeatedly slay the Tempest Lizard in Suhalla Desert, since it is the only boss that respawns an unlimited number of times and yields up to 1768 EXP. It also seems to be highly susceptible to the venom ailment that Bane inflicts, making for a speedy battle if the effect connects. Furthermore, the respawning Psynergy Stone at the entrance to Suhalla Desert makes it possible to heal the party indefinitely, an attractive feature for prolonged level-grinding.
In The Lost Age
The most effective method of training in Golden Sun: The Lost Age is killing the monsters of the Phoenix monster line, which are famous for dishing out large amounts of experience points. If you have a data-enhanced game and have gotten the Lure Cap from Babi after winning Colosso, it will make training quicker.
In Dark Dawn
The most effective way of training in Golden Sun: Dark Dawn is by killing the Tua Warriors that appear at the Apollo Sanctum. Each Tua Warrior gives 13,750 Exp when defeated, and have the additional benefit of having Water of Life as a guaranteed drop. It should be noted that Tua Warriors only appear following the triggering of a certain cutscene at Apollo Sanctum.
Also, felling any monster with a Djinni that is of the element that monster has the lowest resistance to will yield increased experience points. For example, felling a Wonder Bird with a Mercury Djinni will cause you to gain 11208 experience points, rather than the usual 8622.