As big of an Apple fan I am, I have struggled to get AirPrint to work well on a number of "Air print compatible" printers.
Apple's bonjour service (how they get driverless printing to work on IOS devices) is kind of picky, so it is crucial that your router plays nice with it. Most Uverse equipment works from my experience, Dlink is very hit or miss, and my Netgear (WNDR 3700) has been flawless. Google your router and "AirPrint" and see what comes back.
I have a Epson printer, and have had great luck with it. It is an Artisan 730. Epson ( and I am sure others as well, HP does I know) makes an app for IOS to turn any printer into an Airprinter. It is called Iprint, and is available for free in the App store
So before you get an Airprint compatible printer, check your router for compatibility, if it passes pick out an Airprint printer. If it does not get a printer whose manufacturer makes a printing app for IOS.
Good luck, happy printing!!