Is there a good way to think of vanishing cycles and nearby cycles?

In general I don't think there's anything easy about nearby and vanishing cycles. However, I tend to find it enlightening to just consider their topology. Namely, if $f:X \to \Bbb C$ is a function on a complex algebraic (or analytic) variety, then the stalk cohomology of the nearby cycles functor applied to some complex of sheaves $F$ at a point $x \in f^{-1}(0)$ is simply the the cohomology (with respect to $F$) of the Milnor fiber of $f$ at $x$. Given the utility of Milnor fibers in studying the topology of singular spaces, I think this is good motivation for the nearby cycles functor: in some sense it bundles the information of the all the Milnor fibers together into a single sheaf.

Now you can also use lots of neat results about Milnor fibers to actually say things about these stalk cohomology groups and their monodromy. For example, if f has an isolated singularity, then the singular Milnor fiber is homotopic to a bouquet of n-spheres, where n is given by the so-called Milnor number. The monodromy can often be calculated with the help of the Thom-Sebastiani theorem, which says that if $f = g + h$ (where $g,h$ are functions on distinct subvarieties of $X$ such that their sum makes sense as a function on $X$) then the monodromy of the Milnor fiber with respect to $f$ is the tensor product of the monodromies with respect to $g$ and $h$.

If we're also interested in vanishing cycles, then we can give a similarly topological description of the stalk cohomology groups. Letting $i$ denote the inclusion of $f^{-1}(0)$ into $X$, we have the exact triangle with the specialization map $i^*F$ to the nearby cycles of $F$ and the canonical map from the nearby cycles of $F$ to the vanishing cycles of $F$ (although perhaps it shouldn't be called canonical because it's the cone of the specialization map). Anyway, looking at the associated long exact sequence provides a topological description of the vanishing cycles (in particular if $f$ is smooth then the vanishing cycles is zero and we get an isomorphism between $i^*F$ and the nearby cycles of $f$).

I wish I could say more about the kind of recognition principle you're looking for, but the only place that I've encountered nearby cycles is in Springer theory, where the nearby cycles of the adjoint quotient map applied to the constant sheaf is isomorphic to the pushforward of the constant sheaf under the Springer resolution (and hence the monodromy action determines the action of the Weyl group on the cohomology of Springer fibers). Perhaps in this case the thing to notice is that the nilpotent cone is singular and also the fiber over zero of the adjoint quotient map, which makes it seem like a potentially good candidate for nearby cycles arguments?

I also have a lot of difficulty seeing what's going on, and am no expert, so take this with a grain of salt.

Here's one small picture (which you might already know) that I found helpful: Massey's description of "vanishing cycles at angle \theta" (see http://arxiv.org/abs/math/9908107, around page 23) which gives geometric intuition for the passage from X_{\infty} to its universal cover. Namely, restrict your family to the segment where t is in e^{i \theta} [0,\epsilon], i.e., a ray emanating from the origin in the complex plane; then proceed as you did before, except no crazy covers needed, because your base is now contractible. This gives a functor (isomorphic to nearby cycles for any fixed \theta) together with an action of monodromy.

I liked the mentioned article by Massey very much too, but I guess the best descriptions are still Deligne's SGA 7,part II, chapters 13, 14, where he even draws some figures (BTW, my impression from those chapters and some formulations in other articles by Deligne is that he thought about generalisations, apparently never published). I found Wall: "Periods of Integrals and Topology of Algebraic Varieties", Griffiths "summary" and his "Periods III" very helpfull, parts of this motivic book too. Arthur Ogus studies vanishing/nearby cycles in the context of log-geometry. Related to the theme is the local monodromy theorem, about that, monodromy-weight conjecture etc., I found Illusie's article in Asterisque 223 very good.