The solar race and lunar race story is popular among all puranas the solar race contains lord Rama and lunar race on other side lord Krishna . Is here someone who can describe the both side story, their kingdoms etc.
The Solar and Lunar dynasties of Vaivasvata Manavantra (7th Manavantra) are given in
some other Puranas too
1. The Solar Dynasty (or Surya Vansh)
Lord Brahma's son was Marichi, Marichi's son Kashyapa and Kakshyapa's son Vivasvana (Sun god). Vivasvana's son was Vaivasvata (7th or current Manu).
Vaivasvata Manu had no children and he arranged for a sacrifice so that he might have a son. Nine sons were born as a result of this sacrifice. Their names were Ikshvaku, Nabhaga, Dhrishta, Sharyati, Narishyanta, Pramashu, Rishta, Karusha and Prishadhra. Manu also had a daughter named Ila (इला).
Note: Subsequently, thanks to a boon. Ila became a man named Sudyumna. Sudyumna's sons were Utkala, Gaya and Vinatashva. Utkala ruled in Orissa, Gaya in the region that is also called Gaya, and Vinatashva in the west. Or few sources say that earlier Sudyumna (i.e a man) was there but due to curse he became a woman (i.e Ila). When he/she was a woman she married Budha and gave birth to a son which is described in the later part of this answer.
Popular kings of the Ikshvaku Vansh
The complete list of the Ikshvaku Vansh is given in Valmiki Ramayana - Bala Kanda - Sarga 70. Few of them are as below:
Raghu (After him, his lineage is called as Raghavas, hence, Lord Shri Rama is called as Raghav.)
Lord Shri Rama
Note: Valmiki Ramayana mention many kings in between Raghu and Aja including Nahusha and Yayati (who are mentioned in Lunar Dynasty too). But there should not be any doubt as names can be duplicate. However, other sources doesn't depict any king between Raghu and Aja.
2. The Lunar Dynasty (or Chandra or Soma Vansh)
Lord Brahma also had a son named Atri and Atri had a son named Soma or Chandra (Moon god).
Soma performed a rajasuya yajna (royal sacrifice). Having performed the sacrifice, Soma became the ruler of all the worlds. This made him very arrogant and he abducted the sage Brihaspati's wife Tara. This led to a terrible war between the devas and the asuras (the fifth war between the devas and the asuras took place over the abduction of Tara and this came to be known as the Tarakamaya war). Tara was eventually restored to Brihaspati, but Soma and Tara had a son named Budha.
Budha married Ila (daughter of Vaivasvata Manu) and they had a son named Pururava. In that lineage then born kings like Nahusha and Yayati. Yayati and Devayani had two sons, Yadu and Tursavu. Yayati and Sharmishtha had three sons, Druhya, Anu and Puru.
Yadu's descendants came to be known as the Yadavas.
Puru's descendants came to be known as the Pauravas.
Popular kings of Yadavas
Kartavirya Arjuna (He is also known as ShashtrAjuna and was the most skilled of fighters. His greatest deeds were his defeat and imprisonment of Ravana, king of Lanka. He eventually met his death at the hands of Lord Parashurama.)
Madhu (After him, his lineage is called as Madhavas. Hence, Lord Krishna is called as Madhav.)
Vrishni (After him, his lineage is called as Vrishneyas. Hence, Lord Krishna is called as Vrisney. His name is mentioned in Bhagavad Gita 10.37.)
Lord Shri Krishna
Popular kings of Pauravas
Bharat (After him, his lineage is called as Bharat Vansh. His name is mentioned in Bhagavad Gita 4.7.)
Kuru (After him, his lineage is called as Kauravas.)
Pandu (After him, his lineage is called as Pandavas.)
Note: Our country's name Bharat Varsh (the region of Bharat) is said to be after king Bharat's name. But other than above mentioned Bharat, many kings were born earlier whose names were Bharat. One of them was king Bharat who was born in 1st Manavantra in the lineage of Priyvrata (the son of Svayambhu Manu). His name is not given here as these lineage are given from Vaivasvata Manavantra (7th Manavantra).
EDIT [to answer SHebbar@'s follow up Qs]
which Purana say that Budha married Ila? Mahabharata says Ila was both father and mother of Pururavas.
Even Mahabharata say that Ila married Budha / Vudha:
Ila will have Vudha for her husband, and from Vudha will spring Pururavas. From Pururavas will spring Ayu. From Ayu will spring Nahusha, and Nahusha will beget a son named Yayati. ~ Mahabharata: Anusasana Parva
Atri had for [his] son Soma. Soma's son was called Vudha. Vudha had one son, of the splendour of the great Indra, called Pururavas. Pururavas had a son called Ayus. Ayus had for his son Nahusha. Nahusha had for his son Yayati who was a royal sage equal to a celestial. ~ Mahabharata, Book 7: Drona Parva: Jayadratha-Vadha Parva
More details can be found in Vishnu Purana: Part 4: Chapter 1 & 6. The answer Why is Nahusha's last name Manava? explais from Vishnu Purana: Part 4: Chapter 1 regarding how Purúravas was born. Vishnu Purana: Part 4: Chapter 6 explais how Budha was born.