All the APs start from 1st. If you're doing slow advancement and are willing to do heavy modifications you could probably mix a few if you tried, but its assumed that they'd be separate campaigns and characters.
They do have a chronology, especially as regards to the favored starting town of Sandpoint. My group is planning on doing RotR, We Be Goblins, then Jade Regent, for this reason (but as separate campaigns characters DMs).
I think there's a timeline of modules/APs around somewhere, or at least threads on advice for mixes compatible for a single party, don't know where though.
As far as I can tell (because I've wondered about this very same thing since discovering the APs), there is a sort of implied order to some of the APs, but only insofar as that the stories often assume the events of another story happened already. You do NOT need to play Rise of the Runelords in order to play Jade Regent -- it just happens to be that Jade Regent's story assumes the Runelords story already happened. Characters in one AP are not assumed to be adventuring in any other AP.
Moreover, the writers of at least Jade Regent made the connection to any previous APs only very minor and even include notes on the possibility of rearranging the AP order if you so wish.
About the only thing that you might want from Runelords if you're planning to run Jade Regent would be the first book's info on the starting town of Sandpoint. Other than that, everything I've seen seems to indicate that you could simply ignore Runelords even exists and still run Jade Regent without any trouble whatsoever.
The first two replies are completely correct but I'd like to add that if you go through the APs in chronological order you will find a lot of interesting cross-references and even some NPCs or their relatives that show the love paizo is giving this line.
These things are in general minor details that are absolutely not required to enjoy any given AP. Think of it as the cameo appearance of your favorite director in one of his movies, you know it's there but the person next to you might not and still gets to love the movie.
All the APs start at 1st. The very first three that Paizo did, which were back when they were working for Wizards, went to 20th, but from Rise of the Runelord on they go to around 15th (depends on the AP, but I think the lowest ever was 13 and the highest was 17). Remember, this is subjective based on a lot of different factors.
If I remember right the official word is that none of the published APs (one of the upcoming APs breaks this rule) assume that the group has run anything previously. A few of the APs take place in similar areas, and a lot of people refer to things as sequals and what not, but this is all just stuff that fans have said/done, not official. There is a lot of crossover with background characters, but for the most part the APs are each very much self contained.
Yup. I'm actually in a similar boat as you since I am coming into PF with Jade Regent. I look at Runelords with the idea that I may one day run it, but JR is definitely going to come first.