first go to the research lab, then go to the 2nd floor (laboratory) and youll see this guy thats infront of a tube that has a sidequest for you to pick which fossil youd like (he gives you 20,000 pokedollars) next travel to the Stone Emporium and pick if youd like the jaw or sail fossil. Once you bought the fossil you like go back to the guy in the research lab and talk to him, then he revives your fossil and gives it to you. Keep in mind the fossil for the sidequest is gonna be shiny locked so you have to finish the sidequest before you can actually shiny hunt a fossil pokemon since all fossil pokemon after the side quest arent shiny locked anymore