You have to visit the city of Solitude in Western Skyrim, to do that you will have to own Greymoor. Once you are inside the city, head towards the marked building (Antiquarian Circle) in the picture below.
You will find a NPC called Verita Numida inside, talk to her to begin your journey! She will give you some simple tasks to complete in the room, once that is taken care of she will send you downstairs to talk with Gabrielle Benele to introduce you into the basic mechanics.
In the next part of the guide we are going to take a look at all the features that the Antiquities System brings along, there is plenty to learn!