Finding $y$-intercepts is very easy on a TI-83/TI-84 calculator. All you have to do is graph the function and use 2nd $\to$ Trace $\to$ Value $\to$ $x=0$.
However, the same method can't be used for finding the $x$-intercepts. The only way I know of finding the $x$-intercepts on these calculators is to use 2nd $\to$ Trace $\to$ zero/intersect $\to$ and then select the left and right bounds of each intercept. Is there a way to find the $x$-intercepts immediately? Such as making a table that shows all the $x$-values for which $y$ equals zero?
You can also draw the help function $y=0$. Then you can do
2nd$\rightarrow$Trace$\rightarrow$intersect. You will have to select the two curves and do a guess, but I think it's still more convenient than giving left/right bounds.