1. The composition of two Möbius transformations is a Möbius transformation, and so is the inverse.
2. Given three distinct points $z_1, z_2, z_3$ and three distinct points $w_1, w_2, w_3$, there exists a unique Möbius transformation $f : \hat{\mathbb{C}} \mapsto \hat{\mathbb{C}}$ that maps $z_j$ to $w_j , j = 1, 2, 3$:

$$\begin{aligned} f_2^{-1} \circ f_1 \end{aligned}$$

where $f_1$ maps $z_1, z_2, z_3$ to $0, 1, \infty$, and $f_2$ maps $w_1, w_2, w_3$ to $0, 1, \infty$.

For example, what is the Möbius transformation that maps $0$ to $-1, i$ to $0$, and $\infty$ to 1 ?

More at Analysis of a Complex Kind.