Classifying prime ideals of $\mathbb{Z}[i]$
You also need the fact that $\textbf{Z}[i]$ is a principal ideal domain. This means that any number in this domain is in one or more principal ideals. Then you only need to check if the three kinds of numbers you've listed belong in only in ideals generated by the kinds of numbers you've listed.
Do you consider $\langle 0 \rangle$ to be a prime ideal? If you don't, then you already have the three classifications:
- $\langle 1 + i \rangle$ has norm 2 and contains $1 - i$, $-1 - i$ and $-1 - i$. So $\langle 2 \rangle$ is a ramifying ideal just like in $\mathbb Z[\sqrt{-5}]$, $\mathbb Z[\sqrt{-13}]$, $\mathbb Z[\sqrt{-17}]$, etc.
- $\langle a + bi \rangle$ where $a$ and $b$ are purely real integers. Then $(a - bi)(a + bi) = a^2 + b^2$ is a positive purely real prime congruent to 1 modulo 4.
- $\langle p \rangle$ where $|p|$ is a purely real prime congruent to 3 modulo 4. These primes are "inert."
Verify that each Gaussian integer is contained in an ideal from one of these three categories (the integer 0 is contained in all ideals, so it doesn't hurt to not consider $\langle 0 \rangle$ prime).
Also note that most diagrams of the Gaussian primes show $-3$, $3i$ and $-3i$, to name just three contained in a Category 3 prime ideal, as inert primes. And rightly so, because those three numbers are also primes, and not just because people want the diagram to be pretty and symmetrical.