Puranas describe many incarnations of Vishnu, Shiva and Devi. As the question is on incarnations of Vishnu, I will restrict my answer to this aspect only.
After reading about Vedic Literature in Internet (I am not a sanskrit scholar), Valmiki Ramayana (It is the only authoritative text on Ramayana), Mahabharata and Puranas, I concluded that the following might be the order in which the literature came into existence.
(1) Vedic Literature (Only 3 vedas please - Atharvana veda is an addition at a later date) , (2) Srimad Ramayana (3) Mahabharata (4) Puranas. Till Mahabharata period, i.e., around 3102 B.C., the Puranas were not written and Vedic Pantheon were used to be adored.
In Vedic Era, Lord Brahma was Lord of all Lokas, (Lokadhipati), and the Ultimate boon giver, Indra's place is next to him (Lord of Devatas - Devesam) and Vayu, Agni, Varuna, Vishnu, etc are among 33 Gods we have. Vishnu's definitely used to assist Indra and Gods occasionally, but not did not have the status of God of Gods.
Sri Rama was described as the incarnation of Vishnu in Puranas and in the PRAKSHIPTA episode in Srimad Ramayana. However, Srimad Ramayana did not mention about earlier incarnations of Vishnu, except Vamana and Parasurama, which were discussed below.
1) It was PRAJAPATI - Satapatha Brahmana, who took the form of BIG fish, and saved the King, when a great deluge took place. In the later Purana's version, Vishnu was attributed with this act.
2) Srimad Ramayana clearly says (Ayodhya Kanda) that it was Lord Brahma, who took the form of VARAHA and brought out EARTH from Water.
सर्वम् सलिलम् एव आसीत् पृथिवी यत्र निर्मिता |
ततः समभवद् ब्रह्मा स्वयम्भूर् दैवतैः सह || २-११०-३
"All was water only in the beginning" from which element the earth was formed. After that, the self-existent Brahma with all the gods came into existence."
स वराहः ततो भूत्वा प्रोज्जहार वसुंधराम् |
असृजच् च जगत् सर्वम् सह पुत्रैः कृत आत्मभिः || २-११०-४
"Thereafter, that Brahma, assuming the form of boar, caused the earth to rise from water and with his sons of pure soul, created the entire world."
3) Kurma - turtle - Avatar was mentioned in Srimad Ramayana. However, those Sargas are not in conformity with Sage Valmiki's style because Shiva and Vishnu were described in those Sargas as GOD of Gods, which was not Valmiki's style.
4) Hiranyakasipu was mentioned in Srimad Ramayana, but Nrishimha Avatar was not mentioned.
न चापि मम हस्तात्त्वां प्राप्तुमर्हति राघवः || ५-२०-२८
हिरण्यकशिपुः कीर्तिमिन्द्रहस्तगतामिव |
"Like Keerthi got by Hiranyakasipu from Indra's hand, Rama is not suited to get you from my hands."
5) Srimad Ramayana describes Vishnu taking the form of dwarf Vamana - not an incarnation (2) He restored the wealth of Indra on one occasion.
6) According to Puranas, Parasurama was the 6th incarnation of Lord Vishnu. He makes a brief appearance in Srimad Ramayana in penultimate Sargas of Bala Kanda.
His story, in his own words, as described in those penultimate Sargas of Bala Kanda of Srimad Ramayana is as follows:
Vishnu handed over a supreme bow to Sage Riciika, the son of Bhrigu, as a trustworthy trust. That great-resplendent Sage Riciika on his part has handed over this divine bow to his son with unmatchable deeds of religious merit, who is my father Sage Jamadagni.
Adhering to a barbarous mentality Kaartvaviirya Arjuna put my father to death, when the ascetically powerful father of mine has isolated himself from arsenal.
On hearing the unregenerate and highly perfidious murdering of my father, I rancorously extirpated Kshatriya-s as and when they are born, that too not for one time, going around the earth... and on getting the entire earth under my control I performed Vedic-ritual, and at the end of that Vedic-ritual, I gave all that earth to sage Kashyapa, a sage with divine soul and with pious observances, as a ritualistic-generosity... and I am at present on Mt. Mahendra practising ascesis and thus conjoined are the powers of ascesis in me.
On hearing about the breakage of Shiva's longbow, then I promptly came here. Thereby, oh, Rama, wield this supernatural and superlative longbow of Vishnu, which is passed on to me from my forefathers and my father. Keep your fealty to Kshatriya-hood in view, and wield this as you have wielded Shiva's longbow”
The Story of Parasurama also appears in Mahabharata, wherein it was mentioned that Drona obtained all his divine weapons from the former.
Drona heard that the illustrious Brahmana Jamadagnya, Rama the son of Jamadagni, had expressed a desire of giving away all his wealth to Brahmanas. Having heard of Rama's knowledge of arms and of his celestial weapons also, Drona, who set his heart upon them as also upon the knowledge of morality that Rama possessed, set out for the Mahendra mountains.
Arrived at Mahendra, Drona approached Rama the son of Jamadagni, worshipped Rama's feet, and told him that he came there with the desire of obtaining his wealth.'
On hearing him, Rama said that he gave away his gold and whatever other wealth he had to Brahmanas! He further stated that the earth also, to the verge of the sea, decked with towns and cities, as with a garland of flowers, was given unto Kasyapa. And, he added that he was left with his body only and his various valuable weapons and that he was prepared to give either his body or his weapons.
Drona prayed to give him all his weapons together with the mysteries of hurling and recalling them and Rama gave all his weapons to Drona,
If we study these 2 stories in depth, we will find 2 contradictions.
1) If the Parasu Rama the son of Jamadagni extirpated Kshatriya-s many times by going around the earth, then lineage of both Ikshvaku's, to which Dasaratha belonged, and Nimi's, to which Janaka belonged to, must have been broken, as Parasu Rama might have eliminated Kshatriya-s of both lineages.
However, while describing the lineages of Ikshvaku's and Nimi's during the marriage ceremony of Sri Rama and Sita, Sage Valmiki did not mention about breakage of those lineages due to Parasu Rama's adventures.
2) In Mahabharata it was mentioned that Drona obtained all his divine weapons from Parasu Rama.
Mahabharata followed Srimad Ramayana, as mention about Sri Rama can be found in Mahabharata but not about Pandavas in Srimad Ramayana.
So if Parasu Rama distributed all his riches along with entire land on earth to Kashyapa and retired to Mt. Mahendra for practising ascesis, how can he still wait for Drona of Mahabharata era for imparting knowledge in divine weapons and then retire to Mt. Mahendra?
The answer is clear.
Parasu Rama existed only in Mahabharata era but not in Srimad Ramayana era. Some scholars might have inserted at a latter date the story of Parasu Rama confronting Sri Rama, the son of Dasaratha, in Srimad Ramayana to make believe the readers that Rama the son of Jamadagni was holder of Supreme Vaishnava Bow and he was infact the incarnation of Lord Vishnu that came before Sri Rama.
Hence, we can conclude that Parasu Rama was not the incarnation of Lord Vishnu.
7) Sri Rama is not an incarnation of Vishnu. It was discussed at Did King Dasaratha think of performing Putrakamesthi to beget sons, apart from Aswamedha Yagna?
8) Except in Puranas and Sri Bhagavatam, Mahabharata did not say that Sri Krishna was an incarnation of Vishnu.