I've created a user-role "staff" that I've limited permissions for, but I'm having trouble changing the permissions so that the user can see the catalog but not make any changes to those products.
Is this functionality possible in core magento?
There is no out of the box way of doing this. In a vanilla install you have the option to do everything with the products or nothing at all.
You will to extend the ACL, add more rights (for product view, edit, delete, ...) the check in every action if the admin is allowed to perform the specific task.