Please find some specific examples of functions for which you want to find vert. or horiz. asy. and their equations. This is a broad topic.
Very generally, vert. asy. connect only to rational functions; if the function becomes undef. at any particular x-value, that x-value, written as x = c, is the equation of one vertical asy.
Very generally, horiz. asy. pertain to the behavior of functions as x grows increasingly large (and so are often associated with rational functions). To find them, we take limits of the functions, letting x grow large hypothetically, and see what happens to the function. Very often you end up with the equation of a horiz. line, your horiz. asy., which the graph usually (but not always) does not cross.