Babies can not recognize themselves in a mirror until they are at least 15-18 months old.
Your child will probably start doing things for himself sometime after his first birthday. But every child develops at his own pace, and yours may be more interested in learning one set of skills before mastering another. So try not to worry too much if your child's developing differently than other kids the same age.
Advances come fast and furiously by 18 months, and while children will still need lots of help and attention for years to come, most will have the basics of self-care–dressing, washing their hands, feeding themselves, and going to the bathroom on their own – mastered by or soon after their fourth birthday.