Lambda the Ultimate

...Raymond Smulyun's To Mock a Mockingbird. A couple of other introductions would be To Dissect a Mockingbird or to play around you can use Combinatory Logic Tutorial.

(Then there's the collection of arbitrary things on my combinator birds page).

Since you mentioned some familiarity with the lambda-calculus, you might find it interesting that there is a simple, straightforward procedure for transforming lambda-calculus programs to combinatory logic. These lecture notes by Grigore Rosu list the rules on slide 65. Try using the rules to transform some simple lambda-calculus expressions to combinatory logic. The process of working it out is really useful for understanding what you can do with S and K. Plus there's not much to read (if you'd like a definition of how S and K work, look at slide 61).

Example:
lambda x . (f x)
-- rule 1 -->
[x] (f x)
-- rule 3 -->
S ([x] f)([x] x)
-- rule 2b (x != f) -->
S (K f)([x] x)
-- rule 2a (x == x) -->
S (K f)(S K K)

And that's the end result. You can check that it's correct by applying it to an argument. Just write "z" to the right of the lambda-expression, and to the right of the combinatory logic expression, and check that they both end up simplifying to the same thing.

Note that the variable f is still there because I didn't say "lambda f ..." anywhere (f is a "free variable"). The combinatory logic expression gets significantly more complicated if you add multiple levels of lambdas (at least if you directly follow the rules without performing any simplification). This is because you end up having to apply rules 3, 4 and 5 a lot. If you do try an example like that (say, lambda f . lambda x . (f x)), keep in mind that (S K K) is the same thing as ((S K) K). Or in general, (a b c ... y z) means (((...((a b) c)...) y) z). For example:

lambda x . (a b c)
-- rule 1 -->
[x] (a b c)
-- rule 3 -->
S ([x] (a b)) ([x] c)
-- rule 3 -->
S (S ([x] a) ([x] b)) ([x] c)
-- rule 2b (3 times) -->
S (S (K a) (K b)) (K c)

Lambda-Calculus and Combinators An Introduction 2nd Edition (July 31,2008)

Here are four different descriptions of the same structure from the point of view of physics, topology, logic, and computation. The computation point of view is about combinators.

Physics, Topology, Logic and Computation: A Rosetta Stone

If you know some category theory there's some excellent motivation for combinatory logic in Lambek and Scott's "Higher Order Categorical Logic". On page 42 they start with a cartesian closed category and then apply Occam's razor to it, whittling away anything in the structure that seems redundant. The bare bones that you are left with are a version of combinatory logic.

I also think Torsten Grust's notes on writing a SASL compiler are really helpful. It's easy to write a "virtual machine" that does combinatory logic and these notes show how to compile lambda expressions to combinator expressions that can run on such a machine. This also makes it clear that combinatory logic is complete, in the sense that any lambda expression can be rewritten using it.

I used to get some nice comments on (and lots of links to) an old course lab I wrote at Oberlin College on combinators. It's certainly aimed at a pretty low level, although it assumes some knowledge of Scheme. I was surprised to see that it's still available via the Wayback Machine:

Oberlin DRAGN Project Combinator Lab

Please forgive the day-glo colors and the funky web navigation: hey, it was the 1900s, you know? :)