Pediatrician or to a CLEAN and NICE Tattoo/Piercing parlor. We got our daughters' ears pierced this past summer at a tattoo/piercing business (where I got my second one a couple of years ago).
I had read and known to many folks who have gotten the "gun" technique and you do NOT have to be certified to do so - heck you can even buy those gun's at places like ebay. With the piercing parlor those people are certified by the state and their livelihood depends on a good sterile environment. Additionally, the gun rips through the cartilage while a good experienced piercer uses a needle that slices through the cartilage - and yes there is a HUGE difference in pain as most will attest the gun way is much more painful compared to needle.
Make sure that your daughter gets Titanium or medical grade material earrings put in after the piercing. Gold earrings tend to have some nickel in them which causes a reaction and sometimes an infection (remember the old addage of turning your earrings ever few hours or every day - well that does NOT occur with titanium or medical grade material earrings).