Sorry if this is an easy one, I'm a little rusty on my group theory. My first guess was that it's simply the inverse limit of the Aut($\mathbb{Z}/p^i\mathbb{Z})$, with the map when $i\leq j$ given by taking $\sigma\in$ Aut$(\mathbb{Z}/p^j\mathbb{Z})$ to the map $\tilde{\sigma}:\mathbb{Z}/p^i\mathbb{Z}\rightarrow\mathbb{Z}/p^i\mathbb{Z}$ defined by solving $\phi\circ\sigma=\tilde{\sigma}\circ\phi$ where $\phi:\mathbb{Z}/p^j\mathbb{Z}\rightarrow\mathbb{Z}/p^i\mathbb{Z}$ is the reduction map, but that seems too optimistic - I couldn't think of any reason $\tilde{\sigma}$ would be well-defined, much less be an automorphism of $\mathbb{Z}/p^i\mathbb{Z}$.

Also, barring a full answer to my question, I would be interested in whether Aut$(\mathbb{Z}_p)$ is a $p$-group. If not, what can we say about the elements $\sigma\in$ Aut$(\mathbb{Z}_p)$ with order a power of $p$?