According to moviebutter.com, James Cameron said he was deep in the process of writing the screenplays for Avatar 2 and 3, and he was almost at the point where the movie was "writing itself." Either he's done or very close it because Zoe Saldana (via Comingsoon) revealed that they're going to shoot the movies in early 2014.
Yes, the two movies will be shooting back-to-back as previously reported.
"I think that they're doing a part two and a part three and they're probably going to shoot it at the same time," Saldana said. "Because if we just do two and then wait, I'll probably be 45 or 50 by the time we get to part three. It takes a long time, it's a very hard process."
Since Saldana's character is a motion captured CGI creation, it wouldn't matter what age she'll be when the movies are filming, but point taken.
With production beginning in 2014, the earliest we'll see the first sequel is probably late 2015, but summer 2016 seems more likely. The sequels are said to feature some underwater environments.
Sigourney Weaver may show up in the sequel(s), but we'll have to see how her character fits in the story if that is happen because her character was killed off in the first movie.