Well, you do the smart thing, and skip forward a long way. The "Avatar" of the last series - a powerful individual who has the unique ability to master all of the four elements - has lived the rest of his life, and died, meaning he has been reincarnated in a new Avatar, to face new challenges decades after the earlier series was set. This means that you have to cast off all the old characters, and setting, and create a new world, evolved from the old but by neccesity it has to be it's own thing; it has to stand on its own. This is The Legend of Korra.
Originally envisioned as a 12-part one-off miniseries, Korra has since been greenlight for a total of 48 episodes, which is great news in many ways, but also explains why it feels a little rushed in places. The pace of the show is breaktaking at times, often in a good way, as there is little filler, but also that filler is often useful when you need to explain a world and a set of characters to an audience, especially a world as interesting as the one you have here. The show is almost entirely set in Republic City, which reminds me of nothing more than pictures of I've seen of 1930s Shanghai, a distinctly oriental, but still very diverse city with cars and airships alongside temples and spiritualism. Things have clearly moved on since the time of Avatar, and not on is there more technology everywhere but many of the rare and esoteric bending techniques (metal, lightning) are now commonplace.
Socially there is also a pro-bending sport, where teams of three benders (one of each element apart from the still-extremely-rare air-benders) compete in a sort of ritual duelling, and the main plot involves social unrest between the bending elites, and non-bending masses. There is seriously a lot going on, and the world outside Republic City, and much of the history between the end of Avatar and now, is hardly touched on.
But one of the big joys of Avatar was the characters, who may have started out as common tropes but quickly evolve away from them, and Korra definitely carries that spirit onwards. For a start there is Korra herself; headstrong and determined, she's a fantastic central presence to the show that never feels like a "do-over" from Aang, the first series central character. Her gradually assembled team are likewise diverse and interesting, the nominal "non-bender" being filled by a technically proficient girl with a penchant for dressing like a 30s movie star, and who also gets a mention for being the losing point of the series surprisingly non-tedious love triange, which reverses the usual nonsense by it being about the boy not being able to make up his mind who he fancies.
The big link to the past comes in the former of the older generation, the parental figures for the heroes (who aren't their actual parents, although unusually for these things Korra isn't an orphan or unhealthily distant from her parents) who are the intermediate generation between the characters of Avatar and the leads of Korra. Serving both as authority figures for the teen heroes to rub up against, and mentors, they also end up with their own big emotional moments and kick-ass fight scenes before being (inevitably) taken out so the main cast has to sort things out themselves.
So yes, there is a lot going on in 12 half-hour episodes, and at times The Legend of Korra does feel like it needed to sit down, breathe and do a couple of character episodes. But crucially it manages to walk that line between being a sequel to Avatar: The Last Airbender, and being it's own story all the same time. It brings forward everything that was good about it's predecessor - writing, direction, score - but feels distinct and fresh along with it. Really looking forward to the next series in 2013; Go Fire Ferrets!