Note: PLEASE note that the below "confirmations" delivered to you by many Avatar fansites are NOT confirmed as of yet. DO NOT take most of the information here as a fact. Also, DO NOT edit things interpreted as "fact" into this article. Furthermore, do not add in "question is" and speculation debates.
These storyline elements remain unconfirmed.
Aang is featured in the series in the Spirit World, as Roku and past Avatars were to Aang in the Last Airbender. This is speculated even more because in order for a new Avatar to be born, the one prior to them must be dead, and because Roku was the last Avatar before Aang, it is safe to assume Aang will act as Korra's guide throughout the series.
There is also a rumor that there will be flashbacks of the original series. This may be part of what was implied when DiMartino mentioned that there was a "definite link" between the old series and this one.
In The Fortune Teller, Aunt Wu said Katara would have her third great granchild before passing away in her sleep. Since Tenzin is her son, and probably a lot older than Korra, it is possible he married and had other kids, which means more Airbenders. However, Aunt Wu did admit to not believing in predestination, so it has yet to be seen if her predictions are in fact fulfilled through Aang and Katara's lineage.
Any person born into the Air Nomads is given the ability to Airbend. Presumably any children that Aang and Katara have will in fact be Airbenders, and any of their grand children will be Airbenders, so it can be speculated that if Aang was in fact The Last Airbender, any Airbenders will be descendants of Aang and Katara.
Zuko from Avatar: The Last Airbender may still be the reigning Fire Lord in Avatar: Legend of Korra. He will be exactly 86 in the series, so it can be presumed that he married Mai and had children, therefore shaping his son or daughter into becoming the next Fire Lord, though he may not show up, but can be referenced in a flashback.
Flashbacks in the new series may pertain to what the group did after The War, such as Zuko searching for his mother, the rebuilding of the Air Temples, the newly rebuilt Southern Water Tribe, and Ba Sing Se after the Order of the White Lotus saved it from the Fire Nation.
Flashbacks may also give insight on the development of Republic City, since a metropolis so large had to gain more people in more than 70 years.
Republic City could've been a smaller nation in the original Avatar series, but was never mentioned, and grew over time. It could've also had another name, though nothing is confirmed.