Before us is Mishnah 4 in the seventh chapter of Maseches Yoma. This Mishnah describes the closing actions of the Kohen Gadol on Yom Kippur: the order of the immersions, the changes of garments and the final services, until he leaves the Beis HaMikdash and is escorted to his home.
The third immersion and the change into the white garments:
The Kohen Gadol is standing in the golden garments, after the third immersion, and from here on this is the order of his actions:
"Kideish yadav v'raglav" - he washed his hands and feet, as is required before changing garments.
He removed the golden garments.
He went down to the mikvah and immersed.
He came up and dried himself.
They brought him the second set of white garments, and he put them on.
Again he sanctified his hands and feet.
Entering to remove the spoon and the pan:
Afterward the Kohen Gadol enters the Kodesh HaKodashim in order to remove from there the spoon that had held the incense and the pan. This entry is made for this purpose alone.
The fourth immersion and the return to the golden garments:
He sanctified his hands and feet in preparation for removing the white garments.
He removed the white garments.
He went down to the mikvah and immersed.
He came up and dried himself.
They brought him the golden garments, and he put them on.
Again he sanctified his hands and feet.
In the golden garments he enters the Heichal, in order to burn the "ketores shel bein ha'arbayim" - the afternoon incense - and to light the lamps of the Menorah.
The afternoon Tamid:
The Mishnah does not mention the offering of the afternoon Tamid, but according to most of the Rishonim the Kohen Gadol offered it at this stage. There is one opinion which holds that he actually offered it earlier, at the time of the third immersion. Either way, at this point the Kohen Gadol has completed all the services incumbent upon him.
The fifth immersion and the conclusion of the service:
He sanctified his hands and feet - for the last time.
He removed the golden garments.
They brought him his own personal garments, and he put them on.
Escorting the Kohen Gadol and the joy of the day:
Once he had finished his service, the people would escort the Kohen Gadol to his home. And he would make it a festive day with a feast for those close to him, because he had emerged in peace from the Holy - from the holiness of the Kodesh HaKodashim - and had come out unharmed. This was a great joy, a great joy for him.
In summary: In this Mishnah we learned the order of the closing actions of the Kohen Gadol: the third immersion and the change into white garments in order to remove the spoon and the pan from the Kodesh HaKodashim; the immersion that followed and the return to the golden garments for the burning of the afternoon incense and the lighting of the Menorah (and according to most of the Rishonim also for the offering of the afternoon Tamid); and finally the last immersion, dressing in his ordinary clothes, the escort by the people and the great joy over having emerged in peace from the Holy.