It seems likely that if life evolved to the point of civilization on a planet where carbon diamonds were not a scarce luxury resource, but rather a plentiful mineral as accessible as copper or iron ore, the unique characteristics they possess would be useful for practical purposes.
To quote from Wikipedia for the sake of simplicity:
diamond has the highest hardness and thermal conductivity of any bulk material.
I imagine that the hardness of diamonds would be useful to a young civilization for cutting, and probably as a weapon for hunting or combat with other civilizations.
But there would be challenges, as well. Unlike stone, which can be easily shaped or carved into useful shapes for tool use, the hardness of diamonds would likewise make them difficult to craft into workable shapes.
Furthermore, it seems like it would take some time before the usefulness of diamond hardness would become significant. For instance, while arrows and spears tipped with diamond come to mind, they wouldn't be of much greater use than sharpened stone for hunting or combat early on, in terms of ability to penetrate an organism's flesh.
At some point, a civilization would reach the point where the hardness of diamonds could be harnessed for practical use. In advanced modern technology, diamond tipped blades and other tools use even tiny amounts of carbon diamond for practical reasons, and that's even despite the scarcity of the resource.
In this hypothetical world, what kinds of early tools could a civilization devise from the diamonds all around them?
Note: while preparing to post this question I noticed that a similar one has been asked. At the risk of being overruled, I believe it is not a duplicate, because this question is particularly focused on how diamonds would first be used as tools by a civilization coming of age on a planet with diamond abundance.