I've been struggling with parametrizing things in the complex plane.
For example, the circle $|z-1| = 1$ can be parametrized as $z = 1 + e^{i\theta}$. I'm not sure how this was done. I understand how we can now write it as $z = 1+\cos t + i\sin t$. In general, how do I parametrize things in the complex plane?
Consider the equation of a circle, $$ x^2 + y^2 = 1.\tag{1} $$ The parametrization of equation (1) is simply \begin{align} x(t) &= \cos(t)\\ y(t) &= \sin(t) \end{align} for $t\in[0,2\pi)$. From your question, we have $$ \lvert z -1\rvert = \lvert x-1 + iy\rvert = \sqrt{(x-1)^2 + y^2} = 1\tag{2} $$ where I let $z= x+iy$. By squaring both sides of equation (2), we have $$ (x-1)^2 + y^2 = 1 $$ In $\mathbb{R}^2$, this would be a circle with center $(1,0)$ or radius $1$. We have already seen the parametrization of circle with the center at the origin of radius $1$. If we can shift that $x(t)$ to the right by $1$, we will be good to go; that is, \begin{align} x(t) &= 1 + \cos(t)\\ y(t) &= \sin(t) \end{align} Now $z(t) = x(t) + iy(t) = 1 + \cos(t) + i\sin(t) = 1 + e^{it}$.
Your second question was how does one go about parametrizing in the Complex plane.
As example, consider a square with in the Complex plane with vertices $(0, 0), (1, 0), (1, 1), (0, 1)$. The direction will be counter clockwise. \begin{align} \gamma_1: & \text{ will go from } (0,0)\to(1,0)\\ \gamma_2: & \text{ will go from } (1,0)\to(1,1)\\ \gamma_3: & \text{ will go from } (1,1)\to(0,1)\\ \gamma_4: & \text{ will go from } (0,1)\to(0,0) \end{align} At $t=0$, we will start at $(0,0)$, and at $t=1$, we need to be at $(1,0)$, and so and so on for the other points. \begin{alignat}{2} \gamma_1(t) &= t &&\quad 0\leq t\leq 1\\ \gamma_2(t) &= 1 + i(t - 1) &&\quad 1\leq t\leq 2\\ \gamma_3(t) &= (3 - t) + i &&\quad 2\leq t\leq 3\\ \gamma_1(t) &= i(4 - t) &&\quad 3\leq t\leq 4 \end{alignat}