I am trying to make a picture in my head so I can understand and remember the rules.
So if $f \circ g$ is onto, it is onto because the function $f$ maps every element from a set $B$ to a set $C$ (thus $f$ is onto) and if $f \circ g$ is one-to-one then every element from set $A$ is mapping an element of set $B$ (and thus is one-to-one).
If both $f$ and $g$ is onto then $f \circ g$ is onto and if both $f$ and $g$ is one-to-one then $f \circ g$ is one-to-one and if both $f$ and $g$ are bijective then $f \circ g$ is bijective?
If $f \circ g$ is bijective, we can't say anything, but that $f$ is onto and that $g$ is one-to-one?
If $f$ is onto and $g$ is one-to-one, nothing can be said?
If $g$ is one-to-one and $g$ is onto, nothing can be said?