Geralt was unusually resilient during the trial of grasses (the process of turning a human into a Witcher) and was exposed to more mutagens than the average Witcher candidates typically are, a side of effect of being exposed to more then the usual amount of mutagens are also what turned his hair white.

Not really important to the story as to why we follow him but he is sort of a special Witcher since he was exposed to more mutagens then normal. There are other Witcher’s that take more to the mutations than Geralt though, like Eskel, for example. Geralt being exposed to more mutations doesn’t make him like a super Witcher compared to other Witcher’s or anything like that.