Automorphisms of the ring $\Bbb Z[x]$ of polynomials with integer coefficients

I am trying to find the automorphisms of the polynomial ring $\mathbb{Z}[x]$.

So far, I have shown that if $\varphi$ is an automorphism, then $\varphi(n)=n$ for $n \in \mathbb{Z}$, and $\varphi(x)$ is a polynomial of degree one, say $mx+n$. If we consider what gets sent to $x$ by $\varphi$, we see that $m$ must be a unit of $\mathbb{Z}$, so $m=\pm 1$. So I think the automorphisms fix $\mathbb{Z}$ and send the indeterminate variable $x \mapsto x+n$ or $x \mapsto -x+n$ for $n \in \mathbb{Z}$. Is this correct?

Your conclusion is correct: the functions $x \mapsto x + n$, $x \mapsto n - x$ are precisely the ring automorphisms of $\mathbb{Z}[x]$. Note that the inverses to the first type are $x \mapsto x - n$, and maps of the second type are involutions, i.e. are their own inverse.