About learning a single Gaussian in total-variation distance
Essentially, this follows from three facts:


learning a Gaussian in total variation distance $\delta$ is equivalent to learning its two parameters, $\mu,\Sigma$, to (respectively) $\ell_2$ and relative Frobenius norms $O(\delta)$. (Since then the "empirical Gaussian" with the mean and covariance you estimated will be $\delta$-close to the true Gaussian).
learning the mean $\mu$ to $\ell_2$ distance $\delta$ can be done with $O(\frac{n}{\delta^2})$ samples. (This is tight)
learning the covariance to relative Frobenius distance $\delta$ can be done with $O(\frac{n^2}{\delta^2})$ samples. (This is tight)


I suggest you try to prove these yourself. The first one follows from relatively "standard" facts about Gaussians [1], the second two are good exercises.

(If you really want a specific reference, I can try to dig some up.)



[1] Theorem 1.3 in this recent paper is overkill, but will do the job. https://arxiv.org/abs/1810.08693