For Aang, the most difficult bending art to master was the one directly opposed to his native one: Earthbending.
Korra, as a Waterbender, should have theoretically had difficulty mastering Firebending, but we see that she was able to Firebend at a really young age, even before she received any training. However, she had real difficulties learning to Airbend. Why is that?
In the world of Avatar, the bending arts have multiple sources. Each Nation is associated with a particular Bending element; why that association is present has yet to be fully clarified. From observation, however, it appears to have both a genetic element and a personality element.
We see that children always have the element of their parents; when the parents have a different Bending element, the children can inherit theirs from the mother or father, like with Aang and Katara's children, or with Mako and Bolin. This would seem to indicate a genetic predisposition towards Bending a particular element.
At the same time, we see throughout both Legend of Korra and The Last Airbender that Bending draws its power from your personality, or Chi. Airbenders focus on evasion and speed. Waterbenders use adaptability and versatility to power their Bending. Earthbenders use stubbornness and resistance. Firebenders use energy and aggression. This can even be seen in the behaviors of the four Nations on a large scale. The Earth Nation focused on defense, and was hesitant to go on an offensive campaign versus the Fire Nation during the 100 Year War, while the Fire Nation's aggression and energy kept them going strong for the whole conflict.
This is where the idea of jings fits in. As discussed by Aang and Bumi, positive jing is when you are attacking, negative jing is when you are retreating, and neutral jing is when you are waiting and listening. Airbending focuses on negative jing. Firebending focuses on positive jing. Earthbending focuses on neutral jing, and Waterbending focuses on alternating between positive and negative as the need arises.
Aang had a hard time with Earthbending because it was against his personality. He avoided combat at all times, if possible. He had trouble exerting his will against the environment around him, preferring to practice avoidance and mitigation. Waterbending was natural for him, as he was very adaptable to his surroundings. Korra, unlike Aang, does not have an Airbender's personality. She is strong willed, and aggressive. Her quick mastery of Firebending is because she can focus her aggression and inner energy into it. Airbending, however, requires peace and evasion. It's simply not in Korra's personality to avoid conflict.
Korra can Firebend easily because she has a Firebender's heart. Her understanding of positive jing allows her to easily focus her aggression. She has trouble with Airbending because her inner Chi does not line up naturally with what is necessary to Airbend. Her negative jing is too weak. It is the Avatar's role to maintain balance within the universe. That is why it is vital for them to master the four elements. Only when they have fully brought balance to their Chi can they hope to bring balance to the world around them.
Airbending requires the bender to center themselves and fight from within, hence the circular motions used by Tenzin and Aang- the bender is always at the middle , physically and spiritually. Korra couldn't root herself spiritually because her other elements were 'outside' of her- easily pushed Earth, direct fire and flowing water. Air is the only one where you are already required to feel what you are doing at a basic level, whereas with other elements only top masters can actually explore the inner properties of the elements- Toph and Bumi understood Earthbending better than most, for example.
Korra doesn't seem to understand any of her elements, which is why she couldn't master Airbending initially. Aang was a master at 12 but struggled to understand even water in the beginning and was afraid of Firebending until the Dragons taught him. Point is, Aang understood his elements very well- Korra doesn't talk about her bending at all; she didn't learn much from her White Lotus Masters or Katara, it seemed.
Ba Gua Zhang is similar to Tai Chi in that it is *internal*, but it focuses on following the movements of different types of animals
I was going to answer your question but Gabe Willard did an awesome job!
What I would say, to put it a tad differently, is that even though Korra is born into waterbending and that is her original element, that does not mean it is truly her strongest element. All elements represent different mindsets, fighting styles, and personalities, and the people who match their personality to their element are very successful.
But in a case such as the Avatar, when they have 4 elements to choose from, if their personality doesn't best fit their "original" element, such as Korra, may mean that their best element and their "original" element aren't necessarily one and the same.
With Aang we saw his personality was very much that of an air bender, and he definitely fit his cultural heritage. Korra, on the other hand, is not flexible and doesn't have a deep sense of love like a waterbender should (as Iroh defines waterbenders).
Although I think she is really a firebender at heart with her strong will, aggressiveness, and passion, which would mean her own inheritant element should be the hardest for her to learn, but I guess she is also stubborn and unyielding like an earthbender which would make airbending difficult.
To add onto the last reply, we must also remember that Korra was discovered to be the Avatar at a very early age and was therefore introduced and trained in the various bending arts fairly quickly. Her young age and adaptability at that time truly helped her hone those three elements. However, given that airbending only had one master at the time (Tenzin), her education of this element was far more limited. She had only begun her training with Tenzin after the first episode of the season. While she may relate to fire more than her inherent waterbending, she was nurtured into the arts much sooner than Aang was.
Aang was inherently an airbending and mastered that element very quickly because of his life with the nomads and his personality. Then, as we know, he ran away just before he would be trained and escaped into cryosleep for a great period of time. When he was freed from the ice he was already put against the Fire Nation and his remaining education in the bending arts. Because Air was his most powerful and comfortable ally, I believe this is why he is seen using it more often than the others.
That's my two cents. I think that the replies above mine have truly touched upon the heart of the question. I'm happy to give my input. :)
I really have nothing to say after the last three comments... they were outstanding! But.....
To cut it all short, Aang had difficultly mastering earth bending because they were based on listening and patience and earth is opposite of air.
Korra had difficultly mastering airbending because it was based on precision, and speed. Korra is more of aggressive and her bending as she thinks is all about power. She has yet to master airbending because she doesn't know the other purposes of bending. Simple as that. :)
There is also the practical aspect. Water is not always available around to bend (which is why Katara carried a water sack), whereas firebending does not require the element to be present.
