Following up on the excellent MWG diagram in Amstell's answer, the fundamental observation needed is that holding $p$ fixed, $e$ and $v$ are inverses of each other. $e$ tells us the amount we need to spend to get a certain amount of utility $u$, while $v$ tells us the maximum amount of utility we can get from a certain expenditure $w$. Whenever we want to convert from utility to wealth, we use $e$; and whenever we want to convert from wealth to utility, we use $v$.
All the key identities can be derived from this observation. For instance, suppose we want to derive an identity for $\partial v(p,w)/\partial p_i$. We already know the corresponding identity for the expenditure function, $\partial e(p,u)/\partial p_i=h_i(p,u)$. To turn this into an identity for $v$, we substitute $w=e(p,u)$, obtaining $v(p,e(p,u))=u$, and differentiate with respect to $p_i$. The chain rule implies
$$\frac{\partial v(p,e(p,u))}{\partial p_i} + \frac{\partial v(p,e(p,u))}{\partial w}\cdot\frac{\partial e(p,u)}{\partial p_i} =0\\
\Longleftrightarrow \frac{\partial v(p,w)}{\partial p_i} = -\frac{\partial v(p,w)}{\partial w}\cdot x_i(p,w)$$
which, if we divide by $-\partial v/\partial w$ on both sides, becomes Roy's identity.
Or, suppose that we want to derive the Slutsky equation, which gives the relationship between the derivatives of Marshallian and Hicksian demand (decomposing a Marshallian demand change into substitution and income effects). Analogously to above, we can substitute $w=e(p,u)$ into Marshallian demand $x(p,w)$ to obtain $x(p,e(p,u))=h(p,u)$. Then, differentiating with respect to $p_i$ on both sides and applying the chain rule gives
$$\frac{\partial x(p,e(p,u))}{\partial p_i} + \frac{\partial x(p,e(p,u))}{\partial w}\cdot \frac{\partial e(p,u)}{\partial p_i} = \frac{\partial h(p,u)}{\partial p_i}\\
\Longleftrightarrow \frac{\partial x(p,w)}{\partial p_i}=\frac{\partial h(p,u)}{\partial p_i} - \frac{\partial x(p,w)}{\partial w}\cdot x_i(p,w)
$$
In general, I think the heuristic "switch between $w$ and $u$ as needed using $v$ and $e$" allows you to get pretty much everything here. (A similar heuristic is also useful if you ever deal with Frisch demand systems, where marginal utility $\lambda$ plays the same role that $w$ and $u$ do in Marshallian and Hicksian demand systems.)
Of course, there is one other key fact used above, which is $\partial e(p,u)/\partial p_i = h_i(p,u)$, which for $w=e(p,u)$ becomes $\partial e(p,u)/\partial p_i = x_i(p,w)$. This is best viewed, instead, as a direct consequence of the venerable envelope theorem.
($\partial v/\partial p_i$ can also be derived from the slightly more advanced version of the envelope theorem, where constraints as well as the objective are allowed to depend on a parameter. Since varying $p_i$ in the utility maximization problem changes the budget constraint rather than the objective, the envelope theorem says that its effect will depend on the Lagrange multiplier on that constraint, which is the marginal utility $\partial v/\partial w$ of wealth. This is a good intuition for why the expression for $\partial v/\partial p_i$ is more complicated than the expression for $\partial e/\partial p_i$, picking up an extra factor.)
