Let $\mu_k$ and $\nu_k$ denote the $k$th moments of $\mu$ and $\nu$ respectively. 

Write 
$$W_p(\mu,\nu)=\inf\{\|X-Y\|_p\colon X\sim\mu,Y\sim\nu\},$$
where $\|Z\|_p:=(E|Z|^p)^{1/p}$. 

$\newcommand\si\sigma$Then for any random variables $X$ and $Y$ such that $X\sim\mu$ and $Y\sim\nu$ we have 
$$|\mu_2^{1/2}-\nu_2^{1/2}|=|\|X\|_2-\|Y\|_2|\le\|X-Y\|_2$$
and 
$$|\mu_1-\nu_1|=|EX-EY|\le\|X-Y\|_1\le\|X-Y\|_2.$$

So, 
$$|\mu_2^{1/2}-\nu_2^{1/2}|\le W_2(\mu,\nu) \tag{1}\label{1}$$
and 
$$|\mu_1-\nu_1|\le W_2(\mu,\nu).$$

---

Concerning \eqref{1}, note that $|\mu_2-\nu_2|$ cannot be bounded in terms of $W_2(\mu,\nu)$. Indeed, if e.g. $\mu$ and $\nu$ are the centered normal distributions over $\Bbb R$ with respective standard deviations $\si+1$ and $\si$, then, by [Proposition 7][1], $W_2(\mu,\nu)=1$, whereas $|\mu_2-\nu_2|=2\si+1\to\infty$ as $\si\to\infty$.


  [1]: https://projecteuclid.org/journals/michigan-mathematical-journal/volume-31/issue-2/A-class-of-Wasserstein-metrics-for-probability-distributions/10.1307/mmj/1029003026.full