Howard Zinn (August 24, 1922 – January 27, 2010) was an historian and political scientist with a PhD in history from Columbia University who had a long and distinguished career as a professor. As well as his popular works, he published scholarship in peer-reviewed journals. These qualifications suggest that his work will generally be factually accurate in the sense that he will get dates correct, spell names correctly, and be a generally reliable recounter of empirical information; he won't make the sort of errors that a high school student would make. That doesn't mean his interpretations of the events he discusses should be taken as anything more than one thinker's understanding of those events.
Interpretation is not a matter of trustworthiness but of persuasiveness. Zinn has a particular political stance and interprets events through that lens. As he is an influential and original thinker, his ideas are worth serious consideration. One should never, though, "trust" interpretations wholesale, but one should think for oneself, reading a wide range of different interpretations and primary materials, and then basing one's thinking about history on careful and judicious examination and reflection of multiple viewpoints rather than blind trust of some one individual.
A People's History of the United States is a popular textbook rather than a work of scholarship, and thus it tends to generalize more than works written for an audience of specialists. Thus although it makes for an interesting introductory text, that is not the same as meticulous scholarship that attempts authoritative treatment of a specific, narrow issue.