Suppose f(x) = g(x)/(h(x)) = (a_nx^n+a_(n-1)x^(n-1)+..+a_0)/(b_mx^m+b_(m-1)x^(m-1)+...+b_0)
If g(x) and h(x) have some common factor k(x) then let
g_1(x) = g(x)/(k(x)) and h_1(x) = g(x)/(k(x)).
The graph of g_1(x)/(h_1(x)) will be the same as the graph of g(x)/(h(x)),
except that any x where k(x) = 0 is an excluded value.
Assuming g(x) and h(x) have no common factor, then there will be vertical asymptotes wherever h(x) = 0. If a root is not a repeated root (or is repeated an odd number of times) then the limit on one side of the asymptote will be oo and on the other -oo. If the root has even multiplicity then the limit on both sides of the asymptote will be the same: oo or -oo.
If n < m then f(x)->0 as x->+-oo
If n >= m then divide g(x) /(h(x)) to get a polynomial quotient and remainder. The polynomial quotient is the oblique asymptote as x->+-oo.
For example, if f(x) = (x^3 + 3)/(x^2 + 2), then:
f(x) = (x^3+3)/(x^2+2) = (x^3+2x-2x+3)/(x^2+2) = x - (2x-3)/(x^2+2)
So the oblique asymptote of f(x) is y = x
Intercepts with the x axis are where f(x) = 0, which mean where g(x) = 0.
The intercept with the y axis is where x=0, so just substitute x=0 into the equation for f(x) to find f(0) = a_0/b_0
Apart from all this, just pick some x values and calculate f(x) to give you coordinates (x, f(x)) through which the graph must pass.
