Find the first rational number between -2 and -3.

We use the formula for the midpoint of two numbers, $\frac{1}{2}(a+b)$.
Let $a = -2$ and $b = -3$.

$$x_1 = \frac{1}{2}(-2 + (-3))$$ $$x_1 = \frac{1}{2}(-2 - 3)$$ $$x_1 = \frac{1}{2}(-5)$$ $$x_1 = -\frac{5}{2}$$

Now, find a rational number between $-2$ and $-\frac{5}{2}$.
Let $a = -2$ and $b = -\frac{5}{2}$.

$$x_2 = \frac{1}{2}\left(-2 + \left(-\frac{5}{2}\right)\right)$$

To add these, find a common denominator:

$$x_2 = \frac{1}{2}\left(-\frac{4}{2} - \frac{5}{2}\right)$$ $$x_2 = \frac{1}{2}\left(-\frac{9}{2}\right)$$ $$x_2 = -\frac{9}{4}$$

Now, find a rational number between $-\frac{5}{2}$ and $-3$.
Let $a = -\frac{5}{2}$ and $b = -3$.

$$x_3 = \frac{1}{2}\left(-\frac{5}{2} + (-3)\right)$$

To add these, find a common denominator:

$$x_3 = \frac{1}{2}\left(-\frac{5}{2} - \frac{6}{2}\right)$$ $$x_3 = \frac{1}{2}\left(-\frac{11}{2}\right)$$ $$x_3 = -\frac{11}{4}$$

The three rational numbers between $-2$ and $-3$ are $\boxed{-\frac{5}{2}, -\frac{9}{4}, -\frac{11}{4}}$.