Echoing what Urs said, "But you should all be using the nLab more: if you want literature on quasi-categories, look up the references section on quasi-categories! :-) " and in regard to your question in particular, "See the nLab page on simplicial sets for links."
By looking at this nLab page on simplicial sets, in the references section you'll have found (as of this date, July 21, 2011 10:19 pm, EST):
Greg Friedman: An elementary illustrated introduction to simplicial sets
which is chock full of pictures illustrating the geometric ideas underlying the combinatorics of simplicial sets. As Friedman discusses in the introduction on the second and third pages of this paper,
"Here, for the most part, you won't find many complete proofs of theorems, and so these notes will not be completely self-contained. Rather, I try primarily to show by example how the very basic combinatorics, including the definitions, arise out of geometric ideas and to show the geometric ideas underlying the most elementary proofs and properties."
Suffice it to say, those notes may be the end of your search in finding a concise introduction to simplicial sets that also helps develop your geometric intuition and the computation of products. You also might find the references I gave in answer to the question here helpful.