Show that the free product $G ∗ H$ of nontrivial groups $G$ and $H$ has trivial center, and that the only elements of $G ∗ H$ of finite order are the conjugates of finite-order elements of $G$ and $H$.
If $z \in Z(G ∗ H)$, then for all $a \in G ∗ H$ we have that $za=az$. In particular for all $g \in G$ we have that $zg=gz$, thus $zgz^{-1}=g$.
Also for all $h \in H$ we have that $zh=hz$, thus $zhz^{-1}=h$.
I'm trying to show that if $z = z_1\cdots z_k$, then each $z_i$ must belong to $G$ and $H$ and I think that this will imply that the word $z$ must be the empty word?
Any $e\neq z\in G*H$ must be in the reduced form $z=g_1h_1g_2h_2...g_nh_n$ where $g_i\in G, h_i\in H$ such that $g_1\neq e$ or $h_1\neq e$. See https://en.wikipedia.org/wiki/Free_product for the definition of the reduced word.
Now assume that $z\in Z(G*H)$ be in reduced form and $z\neq e$. Take two elements $g\in G$ and $h\in H$ such that $g\neq e$, $h\neq e$. From $hz=zh$ we deduce that $g_1 =e$, from $gz=zg$ we deduce that $h_1 =e$. Contradiction.
Second part of the question: It is obvious that any conjugate of an element of $G$ or $H$ of finite order is of finite order. For the other direction, let $x\in G*H$ be of finite order, say $x^n=e$, $n\geq 2$. Assume that $x$ be in the reduced form $x=gAh$ where $A$ is a reduced non-trivial word (starting with an element from $H$ ending with an element from $G$). But then $x^n$ is in the same form of $x$ and is non-trivial. So $x$ can not be in this form. Similarly, $x$ can not be in the reduced form $hBg$. It must be in the reduced form $g_1Cg_2$ or $h_1Dh_2$ where $C$ and $D$ are non-trivial and reduced. Assume that it is in the form $g_1Cg_2$. Then $x^n=g_1Cg_2g_1C...Cg_2g_1Cg_2=e$. We deduce that $g_2g_1=e$ and $C^n=e$. Similirly, for the form $h_1Dh_2$, we deduce that $D^n=e$. $C$ and $D$ are shorther words. Continuing this process for C and D for new $Cs$ and $D$s, we eventually see that in the end the last C must be in $H$ and the last $D$ must be in $G$ and $x$ is either in the form $zgz^{-1}$ or $zhz^{-1}$ where $z\in G*H$, $g\in G$ and $h\in H$.