What is a set of generators for the multiplicative group of rationals?

You're quite right: $(\mathbb{Q^\times, *})$ is generated by the (positive) primes and $-1$: $$ \mathbb{Q}^\times \cong C_2 \times \bigoplus_p \mathbb Z $$

This is essentially the fundamental theorem of arithmetic.

I assume your list continues on $\{\frac{1}{3},-\frac{1}{3},\frac{1}{5},-\frac{1}{5}, \ldots\}$.

You do indeed give a list of generators; but your list is redundant:

• $1$ is contained in every group; it need not be listed
• $-\frac{1}{2} = (-1) \cdot \frac{1}{2}$, so you only need to include one of each associate; it's traditional to take the positive one in your list and remove the negative one.

Also, it's traditional to list the primes as the generators, rather than their inverses. That is, the typical set of generators one uses for $\mathbb{Q}^\times$ is the set

$$ \{ -1, 2, 3, 5, 7, 11, \ldots \} $$

This is a minimal set of generators; the only relationship between its elements is

$$(-1) = (-1)^{-1}$$