What species of bird? Basically the smaller the bird the earlier that you have to start. For example budgies are almost fully feathered by age 3 weeks, and starting sooner is better. However be sure to pay special attention to how the parents act, you may risk parental aggression toward chicks. Sometimes the parents need to be kept out of the box for longer so that they can calm down.
I personally think it is advisable to remove, weigh, and examine each chick daily and record the weight and anything else you find. You might not NOTICE that a baby hasnt gained weight in 3 days just by looking, but the numbers will tell you every time, and that is a sure fine indication that something is going very wrong and you most likely need to step in in some way. So basically my answer is that I start handling my chicks at day 1, and I would start focusing on "taming" around the time they start to pin.