I am trying to understand how to think about the elements of $H^1(X,\mathcal{O}^*_X)$ I know one way is to think about it as the Picard group via the isomorphism. But when I try to "see" the elements of $H^1(X,\mathcal{O}^*_X)$ I get confused.
We have
$$ 0\to \mathbb{Z}\to \mathcal{O}_X\to \mathcal{O}^*_X\to 0$$
From we get the long exact $$\ldots \to H^1(X,\mathbb{Z})\to H^1(X,\mathcal{O}_X)\to H^1(X,\mathcal{O}^*_X)\to \ldots$$
So $H^1(X,\mathcal{O}^*_X):=\dfrac{\text{Ker}(\phi^1:(\mathcal{O}^*_X(X))^1\to (\mathcal{O}^*_X(X))^2)}{\text{Im}(\phi^0: \mathcal{O}^*_X(X)\to (\mathcal{O}^*_X(X))^1)}$
and the maps $\phi$ are from the resolution of $\mathcal{O}^*_X$ i.e
$$0\to \mathcal{O}^*_X(X)\to (\mathcal{O}^*_X(X))^1\to (\mathcal{O}^*_X(X))^2\to \ldots$$
But I don't know what the sheaf $(\mathcal{O}^*_X(X))^1$, (and so on) might be. So how can I make sense of it ?
Should I just be think elements of $H^1(X,\mathcal{O}^*_X)$ as non-vanishing global sections ?
No because that would be absolutely incorrect to do. Global sections of a sheaf $\mathcal{E}$ are given by $H^0(X,\mathcal{E})$.
Elements of $H^1(X,\mathcal{E})$ are not, inherently, geometric objects. At best they can be represented by geometric objects, in certain cases (like de Rham cohomology). But for a general sheaf that is not the case. So the most concrete way to think about $H^1(X,\mathcal{O}_X^*)$ is to represent its elements by the combinatorial objects which are are used to compute it.
In this case, that means we take an open cover with desirable properties $\{U_\alpha\}$. Then an element $f\in H^1(X,\mathcal{O}_X^*)$ can be represented as follows. For each $U_\alpha\cap U_\beta$, we take an element $f_{\alpha\beta}\in \mathcal{O}_X^*(U_\alpha\cap U_\beta)$ such that $f_{\alpha\beta}=1/f_{\beta\alpha}$. The reason being that this is the condition which guarantees that $\delta f =1$, where $\delta$ is the Čech differential. It is (in the current setting) defined by $$\delta(f)_{\alpha\beta\gamma}=f_{\beta\gamma}f_{\alpha\gamma}^{-1}f_{\alpha\beta}=f_{\alpha\beta}f_{\beta\gamma}f_{\gamma\alpha}$$ You may recognise the latter (equated to $1$) as the "cocycle condition" which the transition functions of vector bundles need to satisfy. That is to say there is a bijective correspondence between holomorphic line bundles on $X$ and $\ker\delta$. Now, when are two such holomorphic line bundles isomorphic? This is when two cocycles $f, g$ are related via $f_{\alpha\beta}=h_\beta h_\alpha^{-1}g_{\alpha\beta}$ for $h_\alpha\in\mathcal{O}_X^*(U_\alpha)$. In other words, if $f=(\delta h)g$. Because of this, the group $H^1(X,\mathcal{O}^*)$ can be identified with the group of isomorphism classes of holomorphic line bundles on $X$, so ultimately in this specific case, an object in this sheaf cohomology group can be represented by something geometric. But that isn't always the case so the combinatorial representation is much more concrete.