The Adventure Paths are all standalone; each takes a new set of characters from level 1 to somewhere in the teens. Some NPCs or relatives of NPCs show up in later APs (like Sandpoint from Rise of the Runelords reappears in Jade Regent, and various Vancaskerkin relatives appear in many of the APs), but that's more of a cool Easter egg than a necessary plot point.
Start with whichever AP you'd like, based on whether you want traditional mixed-field D&D (Rise of the Runelords), horror (Carrion Crown), sandbox empire building (Kingmaker), Asian (Jade Regent), etc. I'd recommend starting with a recent one just so it's easily available and so there's active discussion going on on paizo.com. Also, APs before Council of Thieves were all actually 3.5e rules and not Pathfinder. Rise of the Runelords does fit that category, though - they are re-releasing a Rise of the Runelords Anniversary Edition this month (July 2012) all in one volume, updated to the Pathfinder rules, with lots of other hullaballoo around it. So if I were starting from ground zero, I probably would go with this reissue of the classic personally. (And I own all of them, have played most of them, and have GMed some of them.)