When I asked my pediatrician about our daughter, she said whenever we feel comfortable getting her ears pierced, and she recomended La Lobe on Preston in Dallas. We had our daughter done at 6 months. When I was at La Lobe, I asked them about infants, and they said generally the best age is 3 weeks to 3 months because they pull on them less. We have lost several earing backs from rolling, sleeping, etc. In the beginning, we also had to keep my daughter from rubbing her earings. She's 8 months old now and doing well. In retrospect, I'd probably have them done a little younger, closer to 2 - 3 months.
I know some mothers are concerned with hole movement as they grow older. I talked to the ladies at La Lobe, and between their answers and my own experiece (mine were done when I was an infant), I was comfortable with getting my daughter's ears pierced. I'd recomend talking to your pediatrician, and if you are looking for a place, calling La Lobe with questions about what they do. They were very informative, had verbal and printed care instructions, scheduled a follow-up visit to check on her healing, told us what kind of earings to use on an infant, etc.