The 3-dimensional analogues should be $\mathbb{P}(O(a)\oplus O(b)\oplus O(c))$, yes. These are exactly the $\mathbb{P}^2$-bundles over $\mathbb{P}^1$.
In general, any projectivization of a vector bundle on $\mathbb{P}^1$ is toric. (See the book by Cox, Little, Schenck). If I remember correctly, all smooth toric varieties of picard number 2 is of this form.
The blow-up of $\mathbb{P}^3$ at a point is however not of the $\mathbb{P}(O(a)\oplus O(b) \oplus O(c))$, because the blow-up does not even admit a morphism to $\mathbb{P}^1$.
But $\mathbb{P}(O \oplus O \oplus O(1))$ defines the blow-up of $\mathbb{P}^3$ along a line. (The proof is similar to the surface case).