Is there a way to see who is assigning the tickets and who they are assigning them too?

You can go to the Ticket list view--> Choose created time in the filters--> You will be able to view the agent to whom the ticket is assigned to in the same view. To know how it was assigned click on the ticket--> once inside the ticket details page, click on Activities at the top right corner--> You will be able to track how it was assigned.