After the war Sita is brought to Rama before the army. He tells her he fought to clear his name, not for love of her, and she may go where she likes. She answers him: her heart was always his, and what her body suffered in another's power was not her doing. She asks Lakshmana to build a pyre, walks round Rama, prays to Agni to protect her if she is true, and goes into the flames. The gods appear and Brahma tells Rama he is Narayana and Sita is Lakshmi. Agni rises from the fire holding Sita unharmed and gives her back, pure; Rama says he never doubted her, but the world had to see.
The Ramayan of Valmiki, tr. Ralph T. H. Griffith (Project Gutenberg 24869)
Later tellings take the pain out of it. In the Kurma Purana Agni hides the real Sita before Ravana comes and leaves a double, who is burnt away in the ordeal. In the Adhyatma Ramayana Rama himself asks Sita to live in the fire for a year and leave her shadow in the hut. Tulsidas follows this. Some Puranas give the shadow a life of her own, ending as Draupadi.
Sources: The Ramayan of Valmiki, tr. Ralph T. H. Griffith (Project Gutenberg 24869); Maya Sita (Wikipedia)
Hanuman leaps the ocean to Lanka, passing the mountain Mainaka, the serpent mother Surasa and the shadow-catcher Simhika, searches the city by night, finds Sita in the Ashoka grove and gives her Rama's ring, then lets himself be caught and burns Lanka with his own tail.
At Janaka's court in Mithila no king can lift the bow of Shiva that is Sita's bride-price; the young Rama lifts it, strings it and breaks it in two, and wins Sita. Tulsidas sets it at a crowded contest, and brings in Parashurama, furious at the sound.
On the first night of the exile Rama reaches the Ganga at Shringaverapura, where the Nishada king Guha welcomes him and ferries him across. In Tulsidas the boatman will not let him into the boat until he has washed his feet, in case their dust turns the boat into a woman.
King Janaka, ploughing the ground for a sacrifice, finds a baby girl in the furrow and names her Sita, the furrow; long before, the Rigveda prayed to Sita the furrow herself, for good harvests.
Cast off by Rama because of what the people said, Sita bears twin sons in Valmiki's hermitage; when the boys sing the Ramayana at Rama's sacrifice and she is called to swear her purity again, she asks the Earth to take her, and the ground opens.
Rama worships Shiva as a linga on the island of Rameswaram — before the bridge in the Ramayana, or after Ravana's death to expiate the killing of a brahmin in temple tradition, where Sita's sand linga and Hanuman's linga from Kailasa stand side by side.
Rama wishes to worship Shiva on the island before crossing to Lanka, or after killing Ravana, to be cleansed of killing a brahmin.
Vishnu offers a thousand lotuses and his own eye at Kashi; Rama raises a sand linga at Rameswaram before crossing to Lanka; the full record is I-10 in the interactions file.