A peach is softer than apples or most other fruit, so it important to pick a peach using the sides of your fingers rather than the tips.  If you use the tips of your finger to hold the peach, then you will put little dents in the peach (left picture).  Using the sides of your fingers is more gentle and less likely to cause bruising.  Grab the peach firmly and pull it straight off the branch (right picture)..