In Air Nomad tattoos when he was 12 years old and went into the ice. Of course, being an airbender, it makes sense that Aang would at least master his native element before moving onto anything else - and that next element turned out to be water.
After ing up with Katara and Sokka - and after awakening from his 100-year slumber - Aang and Team Avatar traveled to the Northern Water Kingdom to find him a waterbending teacher, Pakku. Katara taught Aang, too, but it was Pakku who was his official waterbending master. Then Team Avatar traveled through the Earth Kingdom in season 2 to find him an earthbending master - Toph Beifong, who is arguably the most powerful earthbender in the world. And finally, Aang properly learned firebending from Zuko towards the end of season 3. But even though he struggled the most with earthbending, he learned firebending last - why?
One obvious reason would be that the world - and Aang, for that matter - was at war with the Fire Nation, so obtaining a firebending teacher wasn't easy. However, there's a deeper reason behind Aang learning firebending last. According to the Avatar Extras from Book One: Water, particularly notes from the episode "The Deserter", Avatars follow the order of the seasons when they master new elements. Aang started with airbending and then moved to waterbending, earthbending, and firebending - in that specific order.
While the Avatar Extras refer to the order as the order of the seasons, it's actually the Avatar Cycle. The Avatar Spirit transfers from one nation/elemental kingdom to the next whenever the Avatar dies. And the Avatar before Aang was Roku, who hailed from the Fire Nation. So taking that into , it makes sense that, for Aang, the final element he would need to master is fire. For Korra, that element was air, which is why she spent The Legend of Korra season 1 learning airbending from Aang's son, Tenzin, in Republic City.
The next Avatar after Korra would be an earthbender, which means he or she would first learn earthbending and then move onto firebending, airbending, and finally, waterbending. Although the Avatar Cycle broke in Legend of Korra season 2, Korra restored Raava's spirit and brought back the cycle, albeit reset. Still, the order in which Avatars learn all four elements shouldn't change in the future - that is, if Avatar: The Last Airbender's creators choose to make another story.