I have heard from my friend that Kshatriya man can marry Vaishya woman. Is it true? What do our scriptures say on this?
Manu Smriti 3.12. For the first marriage of twice-born men (wives) of equal caste are recommended; but for those who through desire proceed (to marry again) the following females, (chosen) according to the (direct) order (of the castes), are most approved.
Manu Smriti 3.13. It is declared that a Sudra woman alone (can be) the wife of a Sudra, she and one of his own caste (the wives) of a Vaisya, those two and one of his own caste (the wives) of a Kshatriya, those three and one of his own caste (the wives) of a Brahmana.
As per the scriptures, a Kshatriya is allowed to marry more than once. So, for the 1st marriage it is recommended that his wife also belongs to the same caste/Varna.
If he wants to marry again, then he can take wives from the Vaishya caste.
Similar verses are found in other Smritis as well.
For example Vyasa Smriti 2.10-11 state:
(A twice born) one can take a wife who is not of his own caste (Varna), even after marrying, one of his own order (Varna). The son begotten on the wife of one's own caste, does not stand as an Asavarna (of a different caste) son to one under the circumstance. (10)
A Brahmana can marry a Kshatriya or Vaishya girl; a Kshatriya can take a Vaishya wife, and a Vaishya can wed a S'udra's daughter. But the member of an inferior caste can not wed a girl of superior caste. (11)
Therefore, it is allowed (but not that much recommended).
This is what the chapter 113 of Markandeya Purana says in this regard
A brahman who marries wives among all the castes, provided that he marries first a brahman woman, incurs no injury in his brahman-hood.
Likewise a kshatriya who marries first a kshatriya’s daughter, incurs no harm if he marries wives from lower castes.
Thus a vaisya, who marries first a vaisya woman and afterwards a girl born from a sudra family, is not excluded from the vaisya family.
The law is thus declared in order. Brahmans, kshatriya, vaisyas, who do not first marry women of the same caste, fall by marrying women of other castes, O king. Whatever excluded woman a man marries after neglecting union in his own caste, of that woman's caste let him indeed become a participator. ~English translation source
Prince Nabhag (son of King Dishta) married a girl from vaisya family without first marrying within his own kshatriya varna. Thus he was declared a vaisya. (this is explained in the above mentioned same chapter)
"If, then, with all the documents before us, we ask the question, does caste, as we find it in Manu and at the present day, form part of the most ancient religious teaching of the Vedas?, we can answer with a decided 'No'. There is no authority whatever in the hymns of the Veda for the complicated system of castes; no authority for the offensive privileges claimed by the Brahmans; no authority for the degraded positions of the Sudras. There is no law to prohibit the different classes of the people from living together, from eating and drinking together; no law to prohibit the marriage of people belonging to different castes; no law to brand the offspring of such marriages with an indelible stigma."
Max Müller in "Chips from a German Workshop", Vol. II, pp. 305 - 306
Yes. Kshatriya man 'can' marry a Vaishya woman, without his children losing their Varna. Means, their child will continue to be a Kshatriya.
A Kshatriya may take three wives. In two of them (viz., the one taken from his own order and the other that is taken from the order immediately below), he takes birth himself (so that those children are invested with the status of his own order). His third wife being of the Sudra order is regarded as very inferior. The son that he begets upon her comes to be called as an Ugra. [source]
There is no restrictions of number of wives for a man, if married appropriately. So the term "three wives" does not mean 1 + 1 + 1 = 3 wives. It means, Kshatriya 'can' have wives from the 3 primary classes of society except Brahmana.
Note: The identity (not to confuse with the Varna / class) of such child born from Kshatriya man & Vaishya woman is called: "Karna".
Famous examples from Mahabharata are Radheya/Vasusena (famously called Karna) and Yuyutsu (Dhritarashtra's son from a Vaishya woman). Refer this answer.