The tenses of the two verbs should match, so it should be “saw”. “see” is wrong. You may be mishearing people saying “The man came to see him”, which is correct because the infinitive “see” is required in that pattern.
Conceptually, the “base form” is the unmarked/uninflected form of a verb, such as is listed in a dictionary. Conceptually, the base form does not have practical use in sentences, but some verb forms that are used in sentences are identical to the base form. The bare infinitive is always identical to the base form.
The “to-infinitive” is simply the infinitive preceded by the word “to”, as occurs in various patterns. I don’t know what you mean by “INFINITIVE VERB”, as distinct from infinitive generally.