Hrm.
I imagine it'd be handled with feats and skills; benders vary pretty well from one individual to the next, with only some trends based on their element.
The main elemental bending stuff would be through the skills Airbending (Cha, trained only), Earthbending (Wis, trained only), Firebending (Cha, trained only), and Waterbending (Wis, trained only). These would be special skills that, in the Special section at the end of their descriptions, would state "You cannot gain ranks in this skill unless you possess the Airbender feat" (or Earthbender, Firebender, or Waterbender, as appropriate to the skill). Each elemental bending task would be accomplished with a check against a particular DC, based on the task (i.e. creating a small gust with Airbending would be a low DC like 5, but creating a gust to knock over a tree would be at least DC 25 or DC 30, if not higher). Some feats (as noted further down) would grant access to some more potent/complex/advanced bending techniques (like creating full-fledged tornados with Airbending and not having to face an obscenely-high DC for the check to do so).
A feat for gaining access to each type of elemental bending, probably something like:
Airbender (General)
Benefit: You may gain ranks in the Airbending skill just as with any other skill. Airbending is a permanent class skill for you, and you gain 1 free rank in it upon taking this feat. Additionally, you gain +1 on Survival skill checks, as airbenders are keen on travel and self-sufficiency. However, you cannot gain the Earthbender, Firebender, or Waterbender feats, unless you possess the Avatar template.
Earthbender (General)
Benefit: You may gain ranks in the Earthbending skill just as with any other skill. Earthbending is a permanent class skill for you, and you gain 1 free rank in it upon taking this feat. Additionally, you gain +1 on Concentration skill checks, as earthbenders are focused individuals, not easily deterred. However, you cannot gain the Airbender, Firebender, or Waterbender feats, unless you possess the Avatar template.
Firebender (General)
Benefit: You may gain ranks in the Firebending skill just as with any other skill. Firebending is a permanent class skill for you, and you gain 1 free rank in it upon taking this feat. Additionally, you gain +1 on Intimidate skill checks, as firebenders are fierce and combative. However, you cannot gain the Airbender, Earthbender, or Waterbender feats, unless you possess the Avatar template.
Waterbender (General)
Benefit: You may gain ranks in the Waterbending skill just as with any other skill. Waterbending is a permanent class skill for you, and you gain 1 free rank in it upon taking this feat. Additionally, you gain +1 on Sense Motive skill checks, as waterbenders are insightful and understanding. However, you cannot gain the Airbender, Earthbender, or Firebender feats, unless you possess the Avatar template.
There'd probably be a few improvement feats for advancing elemental bending abilities, and gaining access to more complex/difficult forms of bending. Such as an Advanced *element*bender feat for each type, an Expert *element*bender feat for each type, and a Master *element*bender feat for each type. Each of these would also add +1 to checks with the associated bending skill, and maybe also the other linked skill (Intimidate for Firebending, frex).
For example, Advanced Waterbender would probably grant the ability to change ice into water and vice versa, through the Waterbending skill, at least to a limited extent. Master Waterbender might allow crystalling ice out of the very moisture in the air. Expert Firebender would probably grant the ability to use lightning through the Firebending skill, at least to some extent, while Master Firebender might allow some manipulation over magma/lava, and manipulating the heat within living creatures' bodies to some extent. Advanced Airbender would probably allow the creation of minor whirlwinds and globes of wind like Aang uses (sitting on an orb of wind to float around at high speed, but with limited upward lift and maneuverability). Master Airbender would probably allow generating tornados and such, or blanketing areas in rapidly-created fog banks from gathering winds. Expert Earthbender might allow shaping crystal with Earthbending checks, while Master Earthbender may allow metalbending to some extent through the Earthbending skill.
Toph, as a naturally-talented earthbender, would probably have reached 15th-level and gained Master Earthbender by the end of the 2nd season of the show. Whereas Zuko, though a capable firebender and swordsman, probably doesn't even have Advanced Firebender yet, despite being almost as high in level (he was pretty soft as a kid and didn't take firebending or combat study very seriously until after his mother died, whereas Toph was born blind and found herself naturally drawn to earthbending early on, without needing a teacher so much as just practice). Of course Zuko probably has more Fighter feats (being a pure Fighter, with perhaps a few levels of Barbarian because of his temper and his constant chasing of the Avatar), while Toph would probably be an Expert/Fighter/Monk or something. Azula's probably an Aristocrat/Fighter with nearly all the Firebending feats possible ('cept for one or two secret technique feats, and maybe not quite Master Firebender yet).
The Advanced feats would probably require something like 9 ranks in the appropriate elemental-bending skill (thus minimum of 6th character level), while the Expert feats would require 12 ranks (9th level minimum thusly), and the Master feats would require 18 ranks (15th level minimum). Characters could take Skill Focus (Airbending) or the like to give the skills a slight boost, and each would probably have a synergy or two. For instance, 5 ranks in Concentration may give +2 synergy on Earthbending checks, while 5 ranks in Intimidate would give +2 synergy on Firebending checks, 5 ranks in Survival would give +2 synergy on Airbending checks, and 5 ranks in Sense Motive would give +2 synergy on Waterbending checks.
A few more feats could be included for thematic purposes and secret techniques, like Iroh's Lightning Diversion (allow redirecting of lightning with a high-but-not-superhigh-Firebending-DC check, and even divert it to strike an enemy instead with a near-epic-DC Firebending check). A Windbourne Flight feat may be around for Master Airbenders to fly without needing gliders or the like. There'd probably also be an Earthsight feat (or Toph's Earthsight if you like) that could be taken at 1st-level (though maybe having "prerequisites: Earthbender and natural blindness, or Expert Earthbender and Skill Focus (Listen) otherwise"). There could be a few feats like Fire Warrior (prerequisite: Firebender, benefit: Bluff and Intimidate are permanent class skills for you, and you get +2 to Initiative), Air Nomad (prerequisite: Airbender, benefit: Jump and Survival are permanent class skills for you, and you get +1 on Reflex saving throws), Water Guardian (prerequisite: Waterbender, benefit: Heal and Sense Motive are permanent class skills for you, and you get +1 on Will saving throws; note that this should be +2 Will or better if elemental bending is the only significant form of magic in the setting), and Earth Adept (prerequisite: Earthbender, benefit: Balance and Concentration are permanent class skills for you, and you get +1 on Fortitude saving throws).
