You need to learn the shapes of the graphs of different types of function: reciprocal, quadratic, cubic etc. This will allow you to identify the shape of the graph by identifying the type of equation that you have.
As for intersections with the axes, they are found by setting x = 0 or y = 0 and solving for the equation for the other variable.
Asymptotes are usually present in a 'division by 0' situation, where the function has no defined value. By setting a value of the variable in a reciprocal function such that you have a 'division by 0', the asymptotes can be found.
In some functions such as cubic and quartic functions, it can be difficult to know whether the graph is 'rising' or 'falling', and simply finding the intersections will not help you with this. Consider the value of y when x is very large and negative and very large and positive to figure this out .
Learning the general shapes of different types of rational function and finding the asymptotes, intersections and general shape in the way described is usually all you need to know sketch a graph of a rational function.