I found it useful to work out the correspondence between monoids and monads; lots of introductory texts on CT give the example of the category Mon so I won't bore you with an account here.  Thinking about monoids and monads helped me move past element-centric thinking toward the arrow-centric way of thinking in CT.  In particular it's helpful to work out how the mu operator of a monad (which composes arrows) is a kind of abstraction of monoid operators (which combine elements).  What you end up with is the monad as a device you can use to make non-monoidal things behave like monoids - closure, associativity, identity.

For bind, google around for Kleisli Category and Kleisli star.  I think of the latter as a kind of hybrid of a function and a functor, although I'm not entirely sure that's correct.

-Gregg