To quote one source among many, "the general reference for vanishing cycles is [SGA 7] XIII and XV". Is there a more direct way to learn the main principles of this theory (i.e. without the language of derived categories), in particular as it applies to the study of certain integral models of curves?
For the purposes of intuition, let me write an "answer" which is probably close to the way Picard or Lefschetz would have thought about this. Suppose that you have a family of complex 
nonsingular plane cubics $X_t$ degenerating to a nodal cubic $X_0$. It is possible to understand the change in topology rather explicitly.

You can set up a basis
of (real) curves $\alpha_t,\beta_t\in H_1(X_t,\mathbb{Z})$,   so that 
$\beta_t\to \beta_0\in H_1(X_0)$ and $\alpha_t\to 0$
as $t\to 0$. So that $\alpha_t$  literally is a vanishing cycle. There is more.
As you transport these cycles around a loop in the $t$-plane, you end up with
a new basis $T(\alpha_t),T(\beta_t)$. This is related to the old basis by the
Picard-Lefschetz formula
$$T(\alpha_t)=\alpha_t$$
$$T(\beta_t) = \beta_t \pm (\alpha_t\cdot\beta_t)\alpha_t$$
You visualize this by cutting along $\alpha_t$ and giving a twist before regluing.

You can jazz this up in various ways of course, and this is the modern theory of vanishing cycles. In modern notation you'll see  a nearby cycle functor $R\Psi$, which corresponds roughly  to $H^*(X_t) = H_*(X_t)^*$ and a vanishing cycle functor $R\Phi$ which measures
the difference between $H^*(X_t)$ and $H^*(X_0)$.
