The horizontal asymptotes are given by the values of y that you can not take. The easiest way I can think of to do this is to evaluate the inverse relation (which would make the x and y values swap) and evaluate the values that x can not take in the inverse relation.
Axial intercepts: Means what are the y intercepts and x intercepts.
Local maxima and minima (aka critical points) are found where the derivative of your function is 0. If the second derivative is positive at a critical point, the critical point is a minimum. If the second derivative is negative, the critical point is a maximum.
Points of inflexion are where the function changes from increasing slope to decreasing slope or vice versa. In other words, points of inflexion are where the second derivative is 0.
A graph is concave up where the second derivative is positive, and concave down where the second derivative is negative.