The best general, understandable description of how to handle polynomials in general is the relevant chapter of Wilf's "Mathematics for the Physical Sciences" (Dover, 2006). The text is somewhat dated, as it is a reprint of a book from before computers being ubiquous, and there are further developments.
The first thing is to look for rational zeros, if any are found, divide their factors out.
For some special forms of polynomials, you can get the zeros by reducing them to lower degree ones, or with known zeros. Up to degree 4, there are formulas in terms of algebraic operations (addition, multiplication, roots), for degree 5 and above there aren't. Also, if there are multiple zeros, they are common to the polynomial and its derivative, so computing the greatest common divisor of the polynomial and its derivative is a first step.
Even when there are explicit formulas, for degrees 3 and 4 they are unwildely, it is often more useful to get a numerical approximation.