In the beginning of Avatar: The Legend of Korra, we see Korra, as a child, bending water, earth and fire, without training to do that. She was born in the Water Tribe, so isn't she supposed to know only how to bend water? Like Aang, who knew only wind and trained a lot to master the other three elements?
The key I think is 'to master'. Both Aang and Korra knew how to bend all four basic elements from since they were born, but required training to get control over such powers (that's why they are avatars; they know how to bend all elements).
She was born in the water tribe, so, isn't she supposed to know only how to bend water?
So, I would answer no to that. She, just like Aang knows how to bend all the four basic elements.
One possible 'proof' is that Aang used his water bending skills (although never being trained by any water bender) when he got lost in the storm and that was how he got into the iceberg in episode 1 in the first place.
As for why Korra has greater mastery over bending than Aang, there was no specific reason mentioned as far as I remember, but I would say that it's just a talent, just like it can be a talent for someone to know how to play piano without actually getting someone teach them while one can say that other people need a push so they can develop certain skills.
Korra was designed to be Aangs polar opposite. He is calm, peaceful, and prefers to solve things without a fight, he prefers to avoid the enemy rather than counter or bide.
Korra on the other hand, is hot-headed, a fighter, prefers to pulverize an enemy rather than avoid a battle.
Airbending is considered to be a very "spiritual" type of bending, it requires one to be connected to his inner spirit. All airbenders were monks, meditated a lot, etc.
The rest of the bendings are more "physical" bendings.
As part of Aang being the exact opposite of Korra, he was familiar with airbending at the beginning of the series, and she was familiar with everything but airbending.
As to why she was able to bend 3 elements at the age of 4-5, that wasn't explained, but we can assume it has to do with pure talent. (She is very good with the "physical" world, but has no talent with the "spiritual" world. An avatar needs both to be "complete".
It is possible that she had some masters over at the South Pole to teach her. After all, Katara commands quite some respect.