I have several Spear Damage Coils in my inventory, however can't figure out how to apply them to my spear. Do I have to purchase a new spear in order to use them?
Since you are able to obtain spear mods, I can assume you have the Frozen Wilds DLC.
By default, you are not able to use any spear mods you come across. In order to unlock the ability to use spear mods, you must complete the quest A Secret Shared within the Frozen Wilds DLC area. It is one of the first set of quests available within the area, so it is pretty hard to miss.
Be warned though. The recommended level for this area is 30+, so you probably would want to power up before heading towards this area.
Aloy's Spear allows for one modification slot.
You can't purchase a new spear, but
when you reach a specific part in the main story, Sylens' Lance allows for two modification slots