Read the book and set the quest active or, if you're feeling particularly adventurous, set the quest active and read the book.
No need to "read" the book again if you still want to find the same trainers later, just set the quest active again.
Spoiler:  Show"read" is short for "open your inventory; click the Trainers of Cyrodiil (fake) book; select the skill in the menu" :)
Basic and advanced trainers are marked from the beginning, apart from the DLC ones which are marked when they "come into being".
Basic trainers are no longer marked when you're too good to be trained by them.
Advanced trainers are no longer marked when you get their skill's 'master trainer' quest.
Master trainers are marked when you successfully complete their quest.
Any trainer is no longer marked in case of an untimely end.