With respect to the norm topology, the space of Radon measures on a domain $\Omega$ is not seperable. Indeed, for any two distinct points $x,y$ in $\Omega$, the Dirac measures $\delta_x$ and $\delta_y$ (where $\delta_x(f)=f(x)$) satisfy $\|\delta_x-\delta_y\|=2$ since you can always find a compactly supported smooth function $f$ with $f(x)=-f(y)=1=\|f\|_\infty$. Any metric space that contains uncountably many disjoint open balls cannot be seperable. Of course there are many subspaces of Radon measures that are seperable in the norm topology, e.g., as you noted $L^1$ naturally embeds as a subspace and is seperable.

The space of Radon measures on a domain $\Omega$ is seperable in the weak$^*$ topology (This is probably the remark you allude to have read somewhere.) Indeed, consider the countable set $M_Q$ of measures of the form $\sum_{x \in S} a_x \delta_x$ where the coefficients $a_x$ are rational and $S$ runs over finite sets of points with rational coordinates. This $M_Q$ is countable and weak$^*$ dense. Also the embedding of $L^1$ as space of measures with absolutely continuous to Lebesgue measure is dense, and this gives another proof of weak$^*$ seperability.

With respect to the norm topology, the space of Radon measures on a domain $\Omega$ is not separable. Indeed, for any two distinct points $x,y$ in $\Omega$, the Dirac measures $\delta_x$ and $\delta_y$ (where $\delta_x(f)=f(x)$) satisfy $\|\delta_x-\delta_y\|=2$ since you can always find a compactly supported smooth function $f$ with $f(x)=-f(y)=1=\|f\|_\infty$. Any metric space that contains uncountably many disjoint open balls cannot be separable. Of course there are many subspaces of Radon measures that are separable in the norm topology, e.g., as you noted $L^1$ naturally embeds as a subspace and is separable.

The space of Radon measures on a domain $\Omega$ is separable in the weak$^*$ topology. (This is probably the remark you allude to have read somewhere.) Indeed, consider the countable set $M_Q$ of measures of the form $\sum_{x \in S} a_x \delta_x$ where the coefficients $a_x$ are rational and $S$ runs over finite sets of points with rational coordinates. This $M_Q$ is countable and weak$^*$ dense. Also the embedding of $L^1$ as space of measures with absolutely continuous to Lebesgue measure is dense, and this gives another proof of weak$^*$ separability.