Ayodhya's tirtha-sthala (holy spots) like the exact location of Shree Rama janma-bhoomi, Rishi Lomasha ashrama (hermitage), Vasishtha-kunda, Brihaspati-kunda, etc. which are mentioned in ancient scriptures all exist in present-day Ayodhya [Ref: Skanda Purana 2.8.1-10].
God Rama killed Tadaka and they reached Rishi Vishwamitra's Siddhashrama, at present-day Buksar in Bihar [Valmiki Ramayana 1.26-30; ramyatra.com].
In Videha, Rishi Gautama had set his wife Ahalya to stone due to an event from the past. This is present-day Ahalya uddhara sthana in Ahiyari South in Bihar state [Valmiki Ramayana 1.48; ramyatra.com]. There is another Gautama ashrama nearby [Google maps].
During their exile, God Rama, Sita and Lakshmana reach Rishi Bharadwaja’s ashrama (hermitage) at Prayaga where River Ganga meets River Yamuna, present-day Prayagraj in Uttar Pradesh state [Valmiki Ramayana 2.54].
From Chitrakoot, God Rama, Sita and Lakshmana visit the nearby ashrama of Rishi Atri and Anasuyadevi [Valmiki Ramayana 2.117].
God Rama then goes south to the banks of River Godavari [Valmiki Ramayana 3.11-13]. Folklore believes this to be Parnasala in present-day Telangana state near the temple town of Bhadrachalam which is also called Dakshina Ayodhya.
Ravana had molested Rishi Kushadhwaja’s daughter Vedavati in Naimisha-vana near Kurukshetra in present-day Harayana state [Ref: Ramayana 7.17].
Rishi Matanga’s ashrama was at Rishyamukha hills, present-day Hampi in Karnataka state.
Valmiki had done severe tapasya (penances) at Mahakala-vana near present-day Ujjain in Madhya Pradesh state and become Rishi Valmiki [Ref: Skanda Purana 5.1.24, 7.1.278].
Lakshmana took a pregnant Sita in a chariot from Ayodhya up to the banks of River Ganga. They then took a boat to cross over to reach the ashrama where Valmiki and many ascetic families lived [Ramayana 7.56-59]. This is present-day Bithoor near Kanpur, where she delivered her twins. As per folklore, there are other Valmiki ashrama too, identified in present-day Amritsar in Punjab state; Valmikinagar on the Bihar state border with Nepal; Tapovan Hill near Vaidyanath jyotir-linga temple in Deoghar in Jharkhand state; and Thiruvanmiyur in Chennai in Tamil Nadu state.
Rishi Garga’s ashrama were at present-day Patteeswaram village in Thanjavur district of Tamil Nadu state and Dunagiri in Almora district of Uttarakhand state. He suggested a name for God Krishna in Gokul.
Rishi Durvasa's ashrama was on the banks of River Yamuna in Vraja-bhoomi [Gopalatapani Upanishad].
Rishi Sandipani’s ashrama was near Mahakala-vana in Avanti in present-day Ujjain of Madhya Pradesh state [Skanda Purana 5.1.27; Bhagavata Purana 10.45.31].
Rishi Vyasa’s forest ashrama was at Naimisha-vana near Kurukshetra in present-day Haryana state [Mahabharata 15.19].
[Source: 'Essence of the Fifth Veda' by Gaurang Damani pages 3-5, 9, 11, 16, 29, 62, 82, 94, 98, 213].