Believe it or not, horses are never born grey, dapple grey, or fleabitten grey! The gene that is responsible for a grey horse is not actually a color gene, but a color modifying gene. Grey horses are, genetically, another color. They may be black, bay, or chestnut- they can even be dilute colors, pinto, or appaloosa patterned. Regardless of original color, the grey gene will eventually fade the pigment of the horse's coat and cause it to turn grey.
Usually signs of a horse carrying a gray gene are apparent at or soon after birth. Grey foals will be born a different color, but will often have grey "goggles" around their eyes or a spattering of grey hairs on their face. Although all horses will grey at different speeds, most grey horses will have begun obvious signs of greying by the time they are old enough to wean.
Many grey horses will be a dark steel grey by 18 months, start showing dapples on a dark grey coat by 2 and over the course of the next 5 or 6 years the dapples (or rather, the dark hair dividing dapples) will become fainter and fainter until dapples are no longer visible and the horse appears completely white. Many grey horses will remain white but some gray horses will begin to gain flecks of black in their coat- called fleabitten grey- after a few years of appearing white. Generally the amount of fleabitten flecks will increase annually thereafter. Of course these time frames are all estimates, and each horse can grey at varying rates. Some horses will keep dark dapples into their twenties and some grey horses will be snow white by their second birthday. Just enjoy having a horse who's color you'll never grow tired of!
Typically fleabitten is an "advanced" stage of greying- usually seen in horses in their teens and later- but because horses can grey out at very different rates, some horses will go fleabitten years earlier than others. Typically it progresses from original color to dappled grey to white-grey, and then after being white for awhile they begin to develop fleabitten spots.
How do you know if your foal will be grey? Grey is a dominant gene so if it is passed, it will absolutely make the foal turn grey- there is no "recessive grey". However, there is only a 50% chance grey will be passed to the foal if one parent is grey. If neither sire nor dam is grey, there is a 0% chance of having a grey foal, if one parent is grey, there is a 50% chance the offspring will be grey, if both parents are grey, there is a 75% chance of a grey foal and a 25% chance of a homozygous grey foal. A homozygous grey foal will appear to be a typical grey horse, but when bred will produce 100% grey horses regardless of what color it is bred to.