I have two continuous functions $f: X \to Y$ and $g: Y \to X$ such that $f \circ g =id_Y$ and $g \circ f =id_X$. If I induce maps $f_{*}$ and $g_{*}$ everywhere I search tells me that it's clear that $(f \circ g)_{*}= f_{*} \circ g_{*}$.
Does anyone have a simple explicit proof why this is? Sorry if I'm missing something completely obvious.
I don't understand why you need the extra assumptions. As far as I know the induced map in homology is defined as $$f_\ast[x]=[f(x)],$$ from which it trivially follows, by associativity of composition of maps, that $$(f\circ g)_\ast[x]=[(f\circ g)(x)]=[(f(g(x))]=f_\ast[g(x)]=f_\ast\circ g_\ast[x].$$
See page 120 (in the pdf) of Hatcher https://www.math.cornell.edu/~hatcher/AT/AT.pdf