Here it is! The big book for medicine-cats!

Let's start with the ceremonies!

Ceremonies

A medicine-cat takes on an apprentice:

( I'll take Jayfeather and Lillykit/paw as an example)

Jayfeather: Cats of Thunderclan, as you know, I will not be around forever. So it's time I took an apprentice. I have chosen a cat who has shown great interest in healing her Clanmates. Your next medicine cat will be Lillypaw.
Bramblestar: Lillypaw, do you accept the post of apprentice to Jayfeather?
Lillypaw: I do.
Jayfeather: Then at the half-moon, you must travel to the Moonpool to be accepted by StarClan before the other medicine cats.
Bramblestar: The good wishes of all Thunderclan will go with you.
The mentor touches noses with the apprentice, and the Clan greets the new apprentice.

At the Moonpool:
Jayfeather: Lillypaw, is it your wish to enter the mysteries of StarClan as a medicine cat?
Lillypaw: It is.
Jayfeather: Then step forward.
Jayfeather: Warriors of StarClan, I present you with this apprentice. She has chosen the path of a medicine cat. Grant her your wisdom and insight so that she may understand your ways and heal her Clan in accordance with your will.
Lillypaw drinks from the Moonpool. Then the medicine cats and their apprentices lay down and have a dream from StarClan

The apprentive gets his/her full name:

Medicine-cat apprentices have to train longer than warrior apprentices. While the warrior apprentice gets his/her name after 6 moons of training, the miedicine cat has to train 9, or more, moons to get his/her full name and even then the apprentice is still not a full medicine-cat. The apprentice is just then a full medicine-cat when his/her mentor dies or retires.

Jayfeather: I, Jayfeather medicine cat of Thunderclan, call upon my warrior ancestors to look down on this apprentice. She has trained hard to understand the ways of a medicine cat, and with your help she will serve her Clan for many moons.

Jayfeather: Lillypaw, do you promise to uphold the ways of a medicine cat, to stand apart from rivalry between the Clans and to protect all cats equally, even at the cost of your life?

Lillypaw: I do.

Jayfeather: Then by the powers of StarClan I give you your true name as a medicine cat. Lillypaw from this moment you will be known as Lillypetal. StarClan honors your intelligence(knowledge, kindness, strength, bravery, etc), and we welcome you as a full medicine cat of Thunderclan.
Jayfeather rests his muzzle on the apprentice's head, and she licks her mentor's shoulder. The other medicine cats greet her by her new name.