Given that you are assuming that $R$ is reflexive, the only thing that can fail for $R\cup R^{-1}$ to be an equivalence relation is transitivity: you should verify that since $I_A\subseteq R$, then $I_A\subseteq R\cup R^{-1}$; and that $R\cup R^{-1}$ is symmetric for every relation $R$. So the only possible pitfall lies in transitivity.
Now, you are assuming that $R$ itself is transitive. So, how can transitivity fail? Say you have $(a,b),(b,c)\in R\cup R^{-1}$; if $(a,b),(b,c)\in R$, then since we are assuming $R$ is transitive, then $(a,c)\in R\subseteq R\cup R^{-1}$. If $(a,b),(b,c)\in R^{-1}$, then $(c,b),(b,a)\in R$, and again by transitivity we conclude $(c,a)\in R$, hence $(a,c)\in R^{-1}\subseteq R\cup R^{-1}$.
So what's left? What happens if $(a,b)\in R$, and $(b,c)\in R^{-1}$, but we do not have $(a,b)\in R^{-1}$ nor $(b,c)\in R$? Can you construct such an example? What will happen then?